Gr 11 Math June P2 (Hard)

Questions and Answers

What is the name of the form of the straight line equation that is used when the gradient and one point on the line are known?

Slope-Intercept Form

Gradient–Point Form (correct)

Gradient–Intercept Form

Two-Point Form

What is the value of y when x is 0 in the gradient–intercept form of a straight line equation?

mx + c

c (correct)

mx - c

What is the definition of the constant c in the gradient–intercept form of a straight line equation?

c = y_1 + mx_1

c = x_1 + my_1

c = y_1 - mx_1 (correct)

c = x_1 - my_1

What form of the straight line equation is derived from the two-point form?

Gradient–Point Form Signup and view all the answers

What are the two pieces of information required to use the gradient–point form of a straight line equation?

Gradient and one point Signup and view all the answers

What is the two-point form of the straight line equation used for?

Determining the equation of a line given two points Signup and view all the answers

What is the step to derive the gradient–intercept form from the gradient–point form?

Make y the subject of the formula Signup and view all the answers

What is the name of the form of the straight line equation that is used when two points on the line are known?

Two-Point Form Signup and view all the answers

What is the relationship between the gradient of a line and its inclination?

The gradient is the tangent of the inclination. Signup and view all the answers

What is the gradient of a vertical line?

Undefined Signup and view all the answers

If two lines are parallel, what can be said about their gradients?

They are always equal. Signup and view all the answers

What is the formula to find the inclination of a line with a negative gradient?

θ = 180° + tan⁻¹(m) Signup and view all the answers

What is the purpose of the point-slope form of a line?

To find the equation of a line given the gradient and a point. Signup and view all the answers

Why is it important to ensure the equation of a line is in gradient-intercept form?

To directly read off the gradient. Signup and view all the answers

What can be said about the tangent of an acute angle?

It is always positive. Signup and view all the answers

What is the gradient of a horizontal line?

0 Signup and view all the answers

What is the relationship between the inclination of a line and the positive x-axis?

The inclination is the angle formed between the line and the positive x-axis. Signup and view all the answers

What is the purpose of using the CAST diagram in trigonometry?

To determine the quadrant of an angle. Signup and view all the answers

What is the condition for two lines to be parallel?

They have the same gradient but different y-intercepts Signup and view all the answers

If two lines have gradients of 2 and -1/2, what can be concluded about the lines?

They are perpendicular Signup and view all the answers

What is the equation of a line perpendicular to y = 3x + 2 and passes through the point (1, 4)?

y = -1/3x + 10/3 Signup and view all the answers

What is the trigonometric identity for tangent in terms of sine and cosine?

tan θ = sin θ / cos θ Signup and view all the answers

What is the Pythagorean identity in trigonometry?

sin^2 θ + cos^2 θ = 1 Signup and view all the answers

What is a useful tip for proving trigonometric identities?

Change all trigonometric ratios to sine and cosine Signup and view all the answers

What does the square identity simplify to?

sin^2 θ + cos^2 θ = 1 Signup and view all the answers

What is a condition for tan θ to be undefined?

θ = k × 90°, where k is an odd integer Signup and view all the answers

What is the first step in finding the equation of a line perpendicular to a given line?

Identify the gradient of the given line Signup and view all the answers

What is the point-slope form of the equation of a line?

y - y1 = m(x - x1) Signup and view all the answers

What is the value of tan(θ) in terms of sine and cosine functions?

tan(θ) = sin(θ) / cos(θ) Signup and view all the answers

What is the result of the expression sin²(θ) + cos²(θ)?

1 Signup and view all the answers

What is the value of sin(180° - θ) in terms of sine of θ?

sin(θ) Signup and view all the answers

What is the value of cos(180° + θ) in terms of cosine of θ?

-cos(θ) Signup and view all the answers

What is the value of tan(180° + θ) in terms of tangent of θ?

-tan(θ) Signup and view all the answers

What is the expression for sin²(θ) in terms of cosine of θ?

1 - cos²(θ) Signup and view all the answers

What is the expression for cos²(θ) in terms of sine of θ?

1 - sin²(θ) Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It can be used to simplify trigonometric expressions involving angles of the form 90° ± θ, 180° ± θ, and 360° ± θ. Signup and view all the answers

If point P is rotated through 90°, what are the coordinates of P'?

(-y, x) Signup and view all the answers

What is the value of sin(180° + θ)?

-sin(θ) Signup and view all the answers

If points P and P' are symmetrical about the x-axis, what are the coordinates of P'?

(x, -y) Signup and view all the answers

What is the value of cos(360° - θ)?

cos(θ) Signup and view all the answers

What is the value of tan(180° + θ)?

tan(θ) Signup and view all the answers

If points P and P' are symmetrical about the origin, what are the coordinates of P'?

(-x, -y) Signup and view all the answers

What is the value of sin(-θ)?

-sin(θ) Signup and view all the answers

What is the value of cos(90° + θ)?

-cos(θ) Signup and view all the answers

What is the value of tan(90° - θ)?

tan(θ) Signup and view all the answers

What is the value of sin(360° + θ)?

sin(θ) Signup and view all the answers

What is the formula for the surface area of a square pyramid?

b(b + 2h_s) Signup and view all the answers

What is the formula for the volume of a triangular prism?

1/2bhH Signup and view all the answers

If the dimensions of a solid are multiplied by a factor k, how does the volume scale?

k^3 Signup and view all the answers

What is the name of the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

Angle at the Center Theorem Signup and view all the answers

What is the formula for the area of a circle?

πr^2 Signup and view all the answers

What is the name of the line that touches a circle at only one point?

Tangent Signup and view all the answers

What is the formula for the surface area of a right cylinder?

2πr(h + r) Signup and view all the answers

What is the formula for the volume of a sphere?

4/3πr^3 Signup and view all the answers

What is the name of the theorem that states that the perpendicular bisector of a chord passes through the center of the circle?

Perpendicular Bisector of Chord Passes Through Circle Center Signup and view all the answers

What is the formula for the area of a trapezium?

1/2(a + b)h Signup and view all the answers

A line segment from the center of a circle is perpendicular to a chord. What is the effect on the chord?

The chord is bisected. Signup and view all the answers

What is the relationship between the angles subtended by the same arc in a circle?

They are equal. Signup and view all the answers

What is the period of the function y = sin(kθ)?

360°/k Signup and view all the answers

What is the effect of a on the shape of the function y = a sin(θ)?

It increases the amplitude. Signup and view all the answers

What is the effect of q on the graph of the function y = a sin(θ) + q?

It shifts the graph vertically. Signup and view all the answers

What are the x-intercepts of the function y = cos(θ)?

(90°, 0), (270°, 0) Signup and view all the answers

What is the effect of k on the period of the function y = sin(kθ)?

It decreases the period. Signup and view all the answers

What is the range of the function y = sin(θ)?

[-1, 1] Signup and view all the answers

What is the effect of a on the amplitude of the function y = a cos(θ)?

It increases the amplitude. Signup and view all the answers

What is the y-intercept of the function y = cos(θ)?

(0°, 1) Signup and view all the answers

What is the primary purpose of using reduction formulae and co-function rules in trigonometry?

To simplify trigonometric expressions and facilitate evaluation. Signup and view all the answers

What is the first step in solving a trigonometric equation?

Use a calculator to find the reference angle. Signup and view all the answers

What is the formula for the area of a trapezium?

A = (1/2) (a + b) × h Signup and view all the answers

What is the surface area of a prism or cylinder?

The total area of the exposed or outer surfaces. Signup and view all the answers

What is the name of the geometric solid that has a polygon as its base and vertical sides perpendicular to the base?

Right prism Signup and view all the answers

What is the primary purpose of unfolding a prism or cylinder into a net?

To calculate the surface area. Signup and view all the answers

What is the formula for the area of a circle?

A = π r^2 Signup and view all the answers

What is the general approach to solving quadratic trigonometric equations?

Substitute suitable values to find the solutions within the given interval. Signup and view all the answers

What is the formula for the circumference of a circle?

C = 2π r Signup and view all the answers

What is the purpose of using the CAST diagram in trigonometry?

To determine the quadrants where the trigonometric function is positive or negative. Signup and view all the answers

What is the formula to calculate the volume of a triangular prism?

V = (1/2)b x h x H Signup and view all the answers

What is the formula to calculate the surface area of a right cone?

SA = r(r + h) Signup and view all the answers

What is the formula to calculate the volume of a cylinder?

V = r^2 x h Signup and view all the answers

If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new volume?

V_1 = kV Signup and view all the answers

What is the formula to calculate the surface area of a sphere?

SA = 4r^2 Signup and view all the answers

What is the formula to calculate the volume of a right pyramid?

V = (1/3)bh x H Signup and view all the answers

If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new surface area?

A_1 = 2[klh + lb + kbh] Signup and view all the answers

What is the formula to calculate the volume of a square pyramid?

V = (1/3)b^2 x H Signup and view all the answers

What is the formula to calculate the surface area of a triangular pyramid?

SA = (1/2)b(h_b + 3h_s) Signup and view all the answers

If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new surface area when two dimensions are multiplied by k?

A_2 = k^2 * 2(lh + lb + bh) Signup and view all the answers

What is the period of the function y = cos(2θ)?

180° Signup and view all the answers

What is the effect of k on the period of y = tan(kθ)?

The period decreases as k increases. Signup and view all the answers

What is the range of y = tan(θ)?

(-∞, ∞) Signup and view all the answers

What is the effect of p on the graph of y = tan(θ + p)?

The graph shifts to the left by p units. Signup and view all the answers

What is the domain of y = cos(kθ)?

{θ : θ ∈ ℝ} Signup and view all the answers

What is the effect of a on the graph of y = a tan(θ)?

The branches of the graph are steeper. Signup and view all the answers

What is the y-intercept of y = cos(kθ)?

(0, 1) Signup and view all the answers

What is the effect of k on the graph of y = cos(kθ)?

The period decreases as k increases. Signup and view all the answers

What is the domain of y = tan(kθ)?

{θ : θ ∈ ℝ, θ ≠ 90°/k, 270°/k} Signup and view all the answers

What is the asymptote of y = tan(kθ)?

θ = 90°/k Signup and view all the answers

What is the value of sin(180° - θ) in terms of sine of θ?

sin(θ) Signup and view all the answers

What is the condition for tan(θ) to be undefined?

cos(θ) = 0 Signup and view all the answers

What is the expression for cos²(θ) in terms of sine of θ?

1 - sin²(θ) Signup and view all the answers

What is the value of tan(180° + θ) in terms of tangent of θ?

−tan(θ) Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It is used to find the values of trigonometric functions for angles greater than 90°. Signup and view all the answers

What is the value of cos(360° - θ) in terms of cosine of θ?

cos(θ) Signup and view all the answers

What is the proof of the quotient identity tan(θ) = sin(θ) / cos(θ)?

Geometry of the unit circle Signup and view all the answers

What is the result of the expression sin²(θ) + cos²(θ)?

1 Signup and view all the answers

What is the main difference between the two-point form and the gradient-point form of the straight line equation?

The two-point form is used when two points on the line are known, while the gradient-point form is used when the gradient and one point on the line are known. Signup and view all the answers

What is the purpose of rearranging the gradient-point form to make y the subject of the formula?

To derive the gradient-intercept form Signup and view all the answers

What is the relationship between the gradient of a line and its inclination?

The gradient of a line is tan(θ) where θ is the inclination of the line Signup and view all the answers

What can be said about the y-intercept of a line in the gradient-intercept form?

It is the point where the line intersects the y-axis Signup and view all the answers

What is the advantage of using the gradient-intercept form of a straight line equation?

It is easier to find the y-intercept of the line Signup and view all the answers

What is the condition for two lines to be parallel?

They have the same gradient Signup and view all the answers

What is the purpose of using the two-point form of a straight line equation?

To find the equation of a line when two points on the line are known Signup and view all the answers

What can be said about the gradient of a vertical line?

It is undefined Signup and view all the answers

What is the relationship between the gradient of a line and its inclination in the second quadrant?

m = -tan(180° - θ) Signup and view all the answers

What is the equation of a line that passes through the point (2, 3) and is parallel to the line y = 2x + 1?

y = 2x + 5 Signup and view all the answers

What is the value of θ when the gradient of a line is -3/4?

tan⁻¹(-3/4) + 180° Signup and view all the answers

What is the condition for two lines to be parallel in terms of their inclination?

θ₁ = θ₂ Signup and view all the answers

What is the equation of a line that passes through the point (1, 2) and has a gradient of 3/2?

y = 3/2x + 1/2 Signup and view all the answers

What is the gradient of a line that is perpendicular to the line y = 2x + 1 and passes through the origin?

-1/2 Signup and view all the answers

What is the value of θ when the gradient of a line is 2/3?

tan⁻¹(2/3) Signup and view all the answers

What is the equation of a line that is parallel to the line y = 3x - 2 and passes through the point (4, 5)?

y = 3x + 1 Signup and view all the answers

What is the relationship between the gradient of a line and its inclination in the first quadrant?

m = tan(θ) Signup and view all the answers

What is the equation of a line that passes through the point (3, 4) and is parallel to the line y = 2x + 3?

y = 2x + 5 Signup and view all the answers

If two lines are perpendicular, what is the relationship between their gradients?

The product of their gradients is -1 Signup and view all the answers

What is the purpose of the point-slope form of a line?

To find the equation of a line given its slope and one point on the line Signup and view all the answers

What is the trigonometric identity for sin²(θ) + cos²(θ)

1 Signup and view all the answers

What is the condition for tan(θ) to be undefined?

θ = 90° Signup and view all the answers

If two lines have gradients of 3 and -1/3, what can be concluded about the lines?

They are perpendicular Signup and view all the answers

What is the first step in finding the equation of a line perpendicular to a given line?

Identify the gradient of the given line Signup and view all the answers

What is the result of the expression sin(θ)cos(θ) + cos(θ)sin(θ)?

sin(θ)cos(θ) Signup and view all the answers

What is the value of tan(θ) in terms of sine and cosine functions?

sin(θ)/cos(θ) Signup and view all the answers

What is the useful tip for proving trigonometric identities?

Change all trigonometric ratios to sine and cosine Signup and view all the answers

What is the equation of a line perpendicular to y = 2x - 3 and passing through the point (1, 5)?

y = -1/2x + 7 Signup and view all the answers

If points P and P' are symmetrical about the line y = x, what are the coordinates of P'?

(y, x) Signup and view all the answers

What is the value of sin(360° - θ) in terms of sine of θ?

sin θ Signup and view all the answers

If P is rotated through 90°, what are the coordinates of P'?

(-y, x) Signup and view all the answers

What is the value of cos(180° + θ) in terms of cosine of θ?

-cos θ Signup and view all the answers

What is the value of tan(90° + θ) in terms of tangent of θ?

-tan θ Signup and view all the answers

What is the value of sin(90° - θ) in terms of cosine of θ?

cos θ Signup and view all the answers

If points P and P' are symmetrical about the origin, what are the coordinates of P'?

(-x, -y) Signup and view all the answers

What is the value of cos(360° + θ) in terms of cosine of θ?

cos θ Signup and view all the answers

What is the value of tan(180° - θ) in terms of tangent of θ?

-tan θ Signup and view all the answers

What is the value of sin(-θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the purpose of using the CAST diagram in trigonometry?

To determine the quadrants where the trigonometric function is positive or negative. Signup and view all the answers

What is the reduction formula for tan(180° + θ)?

tan(θ) Signup and view all the answers

What is the general solution of the equation sin(θ) = 1/2?

θ = 30° + 360°n Signup and view all the answers

What is the surface area of a rectangular prism with a length of 6 cm, a width of 4 cm, and a height of 3 cm?

78 cm² Signup and view all the answers

What is the formula for the area of a trapezium?

1/2(a + b)h Signup and view all the answers

What is the value of sin(90° + θ) in terms of cosine of θ?

cos(θ) Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It allows us to express trigonometric functions in terms of acute angles. Signup and view all the answers

What is the formula for the area of a circle?

πr² Signup and view all the answers

What is the purpose of unfolding a prism into a net?

To find the surface area of the prism. Signup and view all the answers

What is the value of cos(180° - θ) in terms of cosine of θ?

-cos(θ) Signup and view all the answers

What is the formula for the volume of a cylinder?

πr²h Signup and view all the answers

What is the formula for the surface area of a sphere?

4πr² Signup and view all the answers

What is the effect of scaling the dimensions of a solid by a factor k on its volume?

It scales by k³ Signup and view all the answers

What is the name of the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

Angle at the Center Theorem Signup and view all the answers

What is the formula for the surface area of a triangular pyramid?

bh + 3hs Signup and view all the answers

What is the formula for the volume of a square pyramid?

b²h/3 Signup and view all the answers

What is the effect of a on the shape of the function y = a sin θ?

The amplitude increases but the graph is not reflected. Signup and view all the answers

What is the formula for the area of a trapezium?

h(b + a)/2 Signup and view all the answers

What is the period of the function y = sin kθ?

360°/|k| Signup and view all the answers

What is the formula for the area of a circle?

πr² Signup and view all the answers

What is the formula for the volume of a triangular prism?

bhH/2 Signup and view all the answers

What is the effect of p on the graph of the function y = sin(θ + p)?

The graph shifts to the left by p units. Signup and view all the answers

What is the y-intercept of the function y = cos θ?

(0, 1) Signup and view all the answers

What is the effect of scaling the dimensions of a solid by a factor k on its surface area?

It scales by k² Signup and view all the answers

What is the amplitude of the function y = a sin θ + q?

a Signup and view all the answers

What is the effect of q on the graph of the function y = a sin θ + q?

The graph shifts vertically upwards by q units. Signup and view all the answers

What is the formula for the volume of a right cone?

1/3πr^2h Signup and view all the answers

What is the x-intercept of the function y = sin θ?

(180, 0) Signup and view all the answers

What is the formula for the surface area of a sphere?

4πr^2 Signup and view all the answers

What is the range of the function y = sin kθ?

[-1, 1] Signup and view all the answers

What is the domain of the function y = cos θ?

{θ: θ ∈ ℝ} Signup and view all the answers

If the dimensions of a solid are multiplied by a factor k, how does the surface area scale?

It scales by k^2 Signup and view all the answers

What is the minimum turning point of the function y = cos θ?

(180, -1) Signup and view all the answers

What is the formula for the volume of a triangular pyramid?

1/3bhH Signup and view all the answers

What is the formula for the surface area of a right cone?

πr^2 + πrL Signup and view all the answers

If the dimensions of a solid are multiplied by a factor k, how does the volume scale?

It scales by k^3 Signup and view all the answers

What is the formula for the volume of a rectangular prism?

lwh Signup and view all the answers

What is the formula for the surface area of a square pyramid?

b^2 + 2bh Signup and view all the answers

What is the formula for the volume of a cylinder?

πr^2h Signup and view all the answers

What is the formula for the surface area of a triangular prism?

bh + 2l(b + h) Signup and view all the answers

What is the period of the function y = cos kθ?

360°/|k| Signup and view all the answers

What is the effect of k on the graph of y = cos kθ when k > 1?

The period decreases Signup and view all the answers

What is the y-intercept of the function y = cos kθ?

(0, 1) Signup and view all the answers

What is the effect of p on the graph of y = cos(θ + p) when p > 0?

The graph is shifted to the left by p units Signup and view all the answers

What is the period of the function y = tan kθ?

180°/|k| Signup and view all the answers

What is the effect of k on the graph of y = tan kθ when k > 1?

The period decreases Signup and view all the answers

What is the y-intercept of the function y = tan kθ?

(0, 0) Signup and view all the answers

What is the effect of p on the graph of y = tan(θ + p) when p > 0?

The graph is shifted to the left by p units Signup and view all the answers

What is the range of the function y = tan kθ?

(-∞, ∞) Signup and view all the answers

What is the domain of the function y = tan kθ?

{θ: θ ∈ ℝ, θ ≠ 90°/k, 270°/k} Signup and view all the answers

What is the condition required to use the two-point form of the straight line equation?

Two points on the line are known Signup and view all the answers

What is the formula derived from the gradient–point form of the straight line equation?

y = mx - mx_1 + y_1 Signup and view all the answers

What is the value of y_1 - mx_1 in the gradient–intercept form of the straight line equation?

The y-intercept of the line Signup and view all the answers

What is the purpose of the gradient–point form of the straight line equation?

To find the equation of a line when the gradient and one point are known Signup and view all the answers

What is the relationship between the gradient of a line and the constant c in the gradient–intercept form?

The gradient is not related to the constant c Signup and view all the answers

What is the advantage of using the gradient–intercept form of the straight line equation?

It provides a standard form for the equation of a line Signup and view all the answers

What is the formula derived from the two-point form of the straight line equation?

y - y_1 = m(x - x_1) Signup and view all the answers

What is the condition required to use the gradient–intercept form of the straight line equation?

The y-intercept of the line is known Signup and view all the answers

What is the condition for tan θ to be undefined?

cos θ = 0 Signup and view all the answers

What is the result of the expression sin²(θ) + cos²(θ)?

1 Signup and view all the answers

What is the value of sin(180° - θ) in terms of sine of θ?

sin θ Signup and view all the answers

What is the expression for sin²(θ) in terms of cosine of θ?

1 - cos²(θ) Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It can be used to simplify trigonometric functions Signup and view all the answers

What is the value of tan(180° + θ) in terms of tangent of θ?

-tan θ Signup and view all the answers

What is the relationship between the gradient of a line and its inclination when the angle is acute?

m = tan(θ) Signup and view all the answers

What is the value of cos(180° - θ) in terms of cosine of θ?

-cos θ Signup and view all the answers

What is the value of sin(90° + θ) in terms of cosine of θ?

cos θ Signup and view all the answers

Which form of the equation of a line is most useful when finding the equation of a parallel line?

Gradient-intercept form Signup and view all the answers

If a line has a negative gradient, what can be said about the angle of inclination?

It is an obtuse angle Signup and view all the answers

What is the relationship between the gradient of a line and its y-intercept in the gradient-intercept form?

m is independent of c Signup and view all the answers

What is the purpose of using the CAST diagram in trigonometry when dealing with negative gradients?

To determine the quadrants of the tangent function Signup and view all the answers

What is the condition for two lines to be parallel in terms of their gradients?

m1 = m2 Signup and view all the answers

What is the step to derive the equation of a parallel line from a given line?

Identify the gradient of the given line Signup and view all the answers

What is the trigonometric relationship between the gradient of a line and its inclination when the angle is obtuse?

m = tan(180° + θ) Signup and view all the answers

What is the importance of ensuring the equation of a line is in gradient-intercept form when finding the equation of a parallel line?

To identify the gradient Signup and view all the answers

What is the characteristic of the tangent function in the second quadrant?

It is negative Signup and view all the answers

What is the condition for two lines to be perpendicular in terms of their gradients?

m1 × m2 = -1 Signup and view all the answers

What is the trigonometric identity for sine in terms of cosine?

sinθ = sqrt(1 - cos²θ) Signup and view all the answers

What is the first step in finding the equation of a line perpendicular to a given line?

Identify the gradient of the given line Signup and view all the answers

What is the purpose of the square identity in trigonometry?

To simplify trigonometric expressions Signup and view all the answers

What is the condition for tanθ to be undefined?

θ = 90° Signup and view all the answers

What is the result of the expression sin²(θ) + cos²(θ)?

1 Signup and view all the answers

What is the gradient of a line perpendicular to y = 2x + 3?

-1/2 Signup and view all the answers

What is the equation of a line perpendicular to y = 3x + 2 and passes through the point (2, 5)?

y = -x/3 + 7/3 Signup and view all the answers

What is the trigonometric identity for cosine in terms of sine?

cosθ = sqrt(1 - sin²θ) Signup and view all the answers

What is the purpose of using the quotient identity in trigonometry?

To simplify trigonometric expressions Signup and view all the answers

What is the formula for the surface area of a right cone?

2πr(r + h) Signup and view all the answers

What is the formula for the volume of a cylinder?

πr²h Signup and view all the answers

If the dimensions of a solid are multiplied by a factor k, how does the surface area scale?

k² Signup and view all the answers

What is the formula for the surface area of a right cone?

πr(r + h) Signup and view all the answers

What is the formula for the volume of a sphere?

4/3πr³ Signup and view all the answers

What is the formula for the area of a trapezium?

(a + b)h/2 Signup and view all the answers

Which of the following trigonometric identities is used to express trigonometric functions in terms of acute angles?

Reduction formulae Signup and view all the answers

If the dimensions of a rectangular prism are multiplied by a factor k, how does the surface area change?

It increases by k^2 Signup and view all the answers

What is the first step in solving a trigonometric equation?

Use a calculator to find the reference angle Signup and view all the answers

What is the formula for the volume of a sphere?

4/3πr^3 Signup and view all the answers

What is the formula for the area of a circle?

πr² Signup and view all the answers

If the length of a rectangle is multiplied by a factor k, how does the volume of the rectangular prism change?

It increases by k Signup and view all the answers

What is the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

Angle at the Center is Twice the Angle at the Circumference Signup and view all the answers

What is the formula for the area of a trapezium?

A = ½ (a + b) × h Signup and view all the answers

What is the formula for the surface area of a triangular pyramid?

1/2b(h_b + 3h_s) Signup and view all the answers

What is the name of the geometric solid that has a polygon as its base and vertical sides perpendicular to the base?

Right prism Signup and view all the answers

What is the formula for the area of a parallelogram?

bh Signup and view all the answers

What is the purpose of unfolding a prism or cylinder into a net?

To find the surface area of the prism or cylinder Signup and view all the answers

If the height of a cylinder is multiplied by a factor k, how does the volume change?

It increases by k Signup and view all the answers

What is the formula for the volume of a square pyramid?

b²h/3 Signup and view all the answers

What is the formula for the volume of a triangular prism?

1/2bhH Signup and view all the answers

What is the formula for the area of a triangle?

bh/2 Signup and view all the answers

What is the formula for the area of a circle?

A = πr^2 Signup and view all the answers

If the radius of a sphere is multiplied by a factor k, how does the volume change?

It increases by k^3 Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It is used to express trigonometric functions in terms of acute angles Signup and view all the answers

What is the formula for the surface area of a square pyramid?

b(b + 2h_s) Signup and view all the answers

What is the purpose of using the CAST diagram in trigonometry?

To determine the quadrants where the trigonometric function is positive or negative Signup and view all the answers

If the base of a right pyramid is multiplied by a factor k, how does the surface area change?

It increases by k^2 Signup and view all the answers

What is the formula for the circumference of a circle?

C = 2πr Signup and view all the answers

What is the formula for the area of a triangle?

A = ½ b × h Signup and view all the answers

What is the effect of a > 1 on the graph of y = a sin θ?

The amplitude increases. Signup and view all the answers

What is the period of the graph of y = sin kθ?

360°/k Signup and view all the answers

What is the minimum turning point of the graph of y = cos θ?

(180°, -1) Signup and view all the answers

What is the effect of q on the graph of y = a sin θ + q?

The graph shifts vertically upwards by q units. Signup and view all the answers

What is the domain of the function y = sin kθ?

{θ : θ ∈ ℝ} Signup and view all the answers

What is the range of the function y = a sin θ?

[-1, 1] Signup and view all the answers

What is the effect of p on the graph of y = sin(θ + p)?

The graph shifts horizontally to the left by p units. Signup and view all the answers

What is the x-intercept of the graph of y = sin θ?

(90°, 0) Signup and view all the answers

What is the maximum turning point of the graph of y = cos θ?

(0°, 1) Signup and view all the answers

What is the y-intercept of the graph of y = a cos θ + q?

(0°, a + q) Signup and view all the answers

If points P and P' are symmetrical about the y-axis, what are the coordinates of P'?

(-x, y) Signup and view all the answers

What is the value of sin(360° - θ) in terms of sine of θ?

sin θ Signup and view all the answers

If points P and P' are symmetrical about the line y = x, what are the coordinates of P'?

(y, x) Signup and view all the answers

What is the value of cos(90° + θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the value of tan(90° - θ) in terms of tangent of θ?

tan θ Signup and view all the answers

What is the value of sin(180° + θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the value of cos(360° + θ) in terms of cosine of θ?

cos θ Signup and view all the answers

What is the value of tan(180° - θ) in terms of tangent of θ?

-tan θ Signup and view all the answers

What is the value of sin(-θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the value of cos(90° - θ) in terms of cosine of θ?

sin θ Signup and view all the answers

What is the period of the function y = cos kθ?

360°/|k| Signup and view all the answers

What is the effect of k on the period of y = cos kθ?

For k > 1, the period decreases, and for 0 < k < 1, the period increases. Signup and view all the answers

What is the range of the function y = cos kθ?

(-1, 1) Signup and view all the answers

What is the effect of p on the graph of y = cos(θ + p)?

For p > 0, the graph shifts to the left, and for p < 0, the graph shifts to the right. Signup and view all the answers

What is the period of the function y = tan kθ?

180°/|k| Signup and view all the answers

What is the effect of k on the period of y = tan kθ?

For k > 1, the period decreases, and for 0 < k < 1, the period increases. Signup and view all the answers

What is the domain of the function y = tan kθ?

{θ: θ ∈ ℝ, θ ≠ 90°/k, 270°/k} Signup and view all the answers

What is the effect of p on the graph of y = tan(θ + p)?

For p > 0, the graph shifts to the left, and for p < 0, the graph shifts to the right. Signup and view all the answers

What is the effect of a on the shape of the graph of y = a tan θ + q?

For a > 1, the branches of the graph are steeper, and for 0 < a < 1, the branches are less steep. Signup and view all the answers

What is the range of the function y = tan kθ?

(-∞, ∞) Signup and view all the answers

Which of the following is true about the gradient–point form of a straight line equation?

It is used when the gradient and one point on the line are known. Signup and view all the answers

What is the advantage of expressing a straight line equation in the gradient–intercept form?

It is easier to find the y-intercept of the line. Signup and view all the answers

Given a straight line equation in the form y = mx + c, what can be said about the value of c?

It is the y-intercept of the line. Signup and view all the answers

Which of the following is a step in deriving the gradient–intercept form from the gradient–point form?

Expanding the brackets and making y the subject of the formula. Signup and view all the answers

What is the significance of the two-point form of a straight line equation?

It is used to find the equation of a line given two points on the line. Signup and view all the answers

Which of the following is a characteristic of the gradient–point form of a straight line equation?

It is used when the gradient and one point on the line are known. Signup and view all the answers

What is the relationship between the two-point form and the gradient–point form of a straight line equation?

The gradient–point form is derived from the two-point form. Signup and view all the answers

What is the advantage of expressing a straight line equation in the two-point form?

It is easier to find the equation of a line given two points on the line. Signup and view all the answers

What is the reduction formula for ∇(180° - θ) in terms of tan(θ)?

-tan(θ) Signup and view all the answers

If points P and P' are symmetrical about the y-axis, what are the coordinates of P' if P is (√3, 1)?

(-√3, 1) Signup and view all the answers

What is the result of the expression sin²(θ) + cos²(θ)?

1 Signup and view all the answers

What is the reduction formula for cos(180° + θ) in terms of cos(θ)?

-cos(θ) Signup and view all the answers

What is the value of tan(180° - θ) in terms of tan(θ)?

-tan(θ) Signup and view all the answers

What is the expression for cos²(θ) in terms of sine of θ?

1 - sin²(θ) Signup and view all the answers

What is the reduction formula for sin(180° + θ) in terms of sin(θ)?

-sin(θ) Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It can be used to rewrite trigonometric functions in terms of θ. Signup and view all the answers

If point P is rotated through 90°, what are the coordinates of P'?

(-y, x) Signup and view all the answers

What is the value of cos(360° - θ) in terms of cosine of θ?

cos θ Signup and view all the answers

If points P and P' are symmetrical about the x-axis, what are the coordinates of P'?

(x, -y) Signup and view all the answers

What is the value of sin(180° + θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the value of tan(180° + θ) in terms of tangent of θ?

tan θ Signup and view all the answers

If points P and P' are symmetrical about the origin, what are the coordinates of P'?

(-x, -y) Signup and view all the answers

What is the value of sin(-θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the value of cos(90° + θ) in terms of sine of θ?

-sin θ Signup and view all the answers

What is the value of tan(90° - θ) in terms of tangent of θ?

cot θ Signup and view all the answers

What is the value of sin(360° + θ) in terms of sine of θ?

sin θ Signup and view all the answers

What is the relationship between the gradient of a line and its inclination in the first quadrant?

m = tan θ Signup and view all the answers

If a line has a gradient of 3, what is the angle of inclination in degrees?

arctan(3)° Signup and view all the answers

What is the condition for two lines to be parallel in terms of their gradients?

m₁ = m₂ Signup and view all the answers

What is the equation of a line parallel to y = 2x + 1 and passing through the point (3, 4)?

y = 2x + 5 Signup and view all the answers

If a line has a negative gradient, what is the relationship between its inclination and the positive x-axis?

θ = 180° + tan⁻¹(m) Signup and view all the answers

What is the trigonometric identity for the tangent of an angle in terms of sine and cosine?

tan θ = sin θ / cos θ Signup and view all the answers

What is the purpose of the CAST diagram in trigonometry?

To identify the quadrants of the coordinate plane Signup and view all the answers

What is the relationship between the gradient of a line and its inclination in the second quadrant?

m = -tan θ Signup and view all the answers

What is the equation of a line parallel to y = -3x + 2 and passing through the point (2, 5)?

y = -3x + 7 Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It is used to simplify trigonometric identities Signup and view all the answers

What is the main condition for two lines to be perpendicular?

Their gradients multiply to -1. Signup and view all the answers

What is the formula to derive the equation of a line perpendicular to a given line and passing through a specific point?

y - y1 = m(x - x1) Signup and view all the answers

What is the purpose of the quotient identity in trigonometry?

To find the tangent of an angle in terms of sine and cosine. Signup and view all the answers

What is the condition for tan(θ) to be undefined?

cos(θ) = 0 Signup and view all the answers

What is the result of the expression sin²(θ) + cos²(θ) in trigonometry?

1 Signup and view all the answers

What is the first step in finding the equation of a line perpendicular to a given line?

Identify the gradient of the given line. Signup and view all the answers

What is the relationship between the gradients of two perpendicular lines?

They are negative reciprocals. Signup and view all the answers

What is the purpose of using the square identity in trigonometry?

To simplify expressions involving sine and cosine. Signup and view all the answers

What is the main difference between the point-slope form and the standard form of a line equation?

The gradient is explicit in the point-slope form. Signup and view all the answers

What is the trigonometric identity that relates the tangent of an angle to the sine and cosine of the same angle?

tan(θ) = sin(θ) / cos(θ) Signup and view all the answers

What is the value of sin(68°) in terms of cosine?

cos(22°) Signup and view all the answers

To solve a trigonometric equation, what is the first step?

Use a calculator to find the reference angle. Signup and view all the answers

What is the formula for the area of a trapezium?

1/2(a + b) × h Signup and view all the answers

What is the surface area of a prism or cylinder?

The total area of the exposed or outer surfaces. Signup and view all the answers

What is the characteristic of the reduction formula for trigonometric functions?

It allows you to express trigonometric functions in terms of acute angles. Signup and view all the answers

What is the purpose of using the CAST diagram in trigonometry?

To determine the quadrants where the trigonometric function is positive or negative. Signup and view all the answers

What is the formula for the volume of a cylinder?

πr^2 × h Signup and view all the answers

If no interval is given, what is the general solution of a trigonometric equation?

The solution within the interval [0°; 360°] and adding multiples of the period to each answer. Signup and view all the answers

What is the formula for the area of a circle?

πr^2 Signup and view all the answers

If the dimensions of a rectangular prism are multiplied by a factor k, how does the surface area change?

It increases by a factor of k^2 Signup and view all the answers

What is the formula for the surface area of a right cone?

πr^2 + πr × h Signup and view all the answers

What is the formula for the circumference of a circle?

2πr Signup and view all the answers

What is the formula for the volume of a triangular pyramid?

1/3 × b × h × H Signup and view all the answers

What is the surface area of a rectangular prism?

The total area of the six rectangles. Signup and view all the answers

If the dimensions of a rectangular prism are multiplied by a factor k, how does the volume change?

It increases by a factor of k^3 Signup and view all the answers

What is the formula for the surface area of a sphere?

4πr^2 Signup and view all the answers

What is the formula for the volume of a right pyramid?

1/3 × b^2 × H Signup and view all the answers

If the dimensions of a cylinder are multiplied by a factor k, how does the volume change?

It increases by a factor of k^3 Signup and view all the answers

What is the formula for the surface area of a triangular pyramid?

1/2 × b × h_b + 3 × 1/2 × b × h_s Signup and view all the answers

What is the formula for the volume of a sphere?

4/3 × πr^3 Signup and view all the answers

If a line segment from the center of a circle is perpendicular to a chord, what will it do to the chord?

Bisect the chord Signup and view all the answers

What is the period of the function y = sin(kθ)?

360°/|k| Signup and view all the answers

What is the effect of q on the graph of y = a sin(θ) + q?

Vertical shift upwards Signup and view all the answers

What is the amplitude of the function y = a sin(θ)?

|a| Signup and view all the answers

What is the range of the function y = sin(θ)?

[-1, 1] Signup and view all the answers

What is the x-intercept of the function y = sin(θ)?

(180°, 0) Signup and view all the answers

What is the effect of a on the graph of y = a sin(θ)?

Change in amplitude Signup and view all the answers

What is the equation of the axis of symmetry of the function y = sin(θ)?

θ = 90° Signup and view all the answers

What is the domain of the function y = sin(θ)?

All real numbers Signup and view all the answers

What is the y-intercept of the function y = cos(θ)?

(0°, 1) Signup and view all the answers

What is the formula for the surface area of a right cone?

$\pi r (r + h_s)$ Signup and view all the answers

What is the effect of multiplying the dimensions of a rectangular prism by a factor of 2?

The surface area is quadrupled, and the volume is octupled. Signup and view all the answers

What is the formula for the volume of a sphere?

$rac{4}{3} \pi r^3$ Signup and view all the answers

What is the name of the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

Angle at the Center Theorem Signup and view all the answers

What is the formula for the area of a trapezium?

$rac{1}{2} (a + b) * h$ Signup and view all the answers

What is the name of the line that touches a circle at only one point on the circumference?

Tangent Signup and view all the answers

What is the formula for the surface area of a triangular pyramid?

$rac{1}{2} b (h_b + 3h_s)$ Signup and view all the answers

What is the effect of multiplying the dimensions of a cylinder by a factor of 3?

The surface area is tripled, and the volume is sextupled. Signup and view all the answers

What is the formula for the volume of a rectangular prism?

$l * b * h$ Signup and view all the answers

What is the formula for the area of a circle?

$\pi r^2$ Signup and view all the answers

What is the period of y = cos(kθ) if k = 2?

90°/2 Signup and view all the answers

What is the range of y = tan(θ)?

(-∞, ∞) Signup and view all the answers

What is the effect of k on the period of y = tan(kθ)?

The period decreases as k increases Signup and view all the answers

What is the y-intercept of y = tan(kθ)?

(0, 0) Signup and view all the answers

What is the effect of q on the graph of y = a tan(θ) + q?

The graph shifts vertically by q units Signup and view all the answers

What is the domain of one branch of y = tan(kθ)?

{θ : -90°/k < θ < 90°/k, θ ∈ ℝ} Signup and view all the answers

What is the effect of a on the graph of y = a tan(θ) + q?

The branches of the graph are steeper for a > 1 Signup and view all the answers

What is the effect of p on the graph of y = tan(θ + p)?

The graph shifts horizontally by p units Signup and view all the answers

What is the period of y = cos(θ) if the period of y = cos(kθ) is 120°?

360°/k Signup and view all the answers

What is the asymptote of y = tan(kθ)?

The lines θ = 90°/k and θ = 270°/k Signup and view all the answers

Study Notes

Here are the study notes for the provided text:

Equation of a Line*

Two-Point Form

The equation of a straight line can be derived from two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2)
Formula: y−y1x−x1=y2−y1x2−x1\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}x−x1y−y1=x2−x1y2−y1

Gradient-Point Form

Derived from the definition of gradient and the two-point form
Formula: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1=m(x−x1)
Requires the gradient of the line and the coordinates of one point on the line

Gradient-Intercept Form

Derived from the gradient-point form
Formula: y=mx+cy = mx + cy=mx+c
ccc is the y-intercept of the straight line
mmm is the gradient of the line
Inclination of a Line*

Relationship between Gradient and Inclination

Gradient mmm of a line is equal to the tangent of the angle θ\thetaθ it makes with the positive x-axis
Formula: m=tan⁡θm = \tan \thetam=tanθ for 0∘≤θ<180∘0^\circ \leq \theta < 180^\circ0∘≤θ<180∘

Special Cases

Vertical lines: θ=90∘\theta = 90^\circθ=90∘, mmm is undefined
Horizontal lines: θ=0∘\theta = 0^\circθ=0∘, m=0m = 0m=0
Lines with negative gradients: m<0m < 0m<0, tan⁡θ<0\tan \theta < 0tanθ<0
Parallel Lines*

Gradient Relationship

Two lines are parallel if and only if their gradients are equal
Formula: m1=m2m_1 = m_2m1=m2

Finding the Equation of a Parallel Line

Identify the gradient of the given line
Use the point-slope form of the equation of a line
Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
Perpendicular Lines*

Gradient Relationship

Two lines are perpendicular if and only if the product of their gradients is equal to -1
Formula: m1×m2=−1m_1 \times m_2 = -1m1×m2=−1

Finding the Equation of a Perpendicular Line

Identify the gradient of the given line
Calculate the perpendicular gradient
Use the point-slope form of the equation of a line
Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
Trigonometric Identities*

Quotient Identity

Formula: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ
Defined for all values of θ\thetaθ except where cos⁡θ=0\cos \theta = 0cosθ=0

Square Identity

Formula: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
Alternative forms: sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \thetasin2θ=1−cos2θ, cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \thetacos2θ=1−sin2θ
Reduction Formulae*

Angle Sum and Difference Formulae

Formulae for sine, cosine, and tangent of 90∘±θ90^\circ \pm \theta90∘±θ, 180∘±θ180^\circ \pm \theta180∘±θ, and 360∘±θ360^\circ \pm \theta360∘±θ
Reduction formulae allow for the expression of trigonometric functions in terms of acute angles, facilitating simplification and evaluation
Trigonometric Equations*

Solving Trigonometric Equations

Use a calculator to find the reference angle
Use the CAST diagram to determine the quadrants
Use reduction formulae to find the values of the angle
Check the solution using a calculator
Area of a Polygon*

Types of Polygons

Square: Area=s2Area = s^2Area=s2
Rectangle: Area=b×hArea = b \times hArea=b×h
Triangle: Area=12b×hArea = \frac{1}{2} b \times hArea=21b×h
Trapezium: Area=12(a+b)×hArea = \frac{1}{2} (a + b) \times hArea=21(a+b)×h
Parallelogram: Area=b×hArea = b \times hArea=b×h
Circle: Area=πr2Area = \pi r^2Area=πr2, Circumference = 2πr2\pi r2πr
Right Prisms and Cylinders*

Surface Area

Calculate the area of each face and add them together
Formulas for surface area of rectangular prism, cube, triangular prism, and cylinder

Volume

Formula: Volume=Area×HeightVolume = Area \times HeightVolume=Area×Height
Formulas for volume of rectangular prism, cube, triangular prism, and cylinder### Rectangular Prism, Triangular Prism, and Cylinder
Volume of a rectangular prism: l × b × h
Volume of a triangular prism: (1/2)b × h × H
Volume of a cylinder: πr^2 × h

Right Pyramids, Right Cones, and Spheres

Definition of a pyramid: a geometric solid with a polygon base and sides that converge at a point (apex)
Definition of a right pyramid: a pyramid with a line from the apex to the center of the base perpendicular to the base
Definition of a cone: a geometric solid with a circular base and sides that converge at a point (apex)
Definition of a sphere: a perfectly round solid, looking the same from any direction

Surface Area and Volume of Pyramids, Cones, and Spheres

Surface area of a square pyramid: b(b + 2hs)
Surface area of a triangular pyramid: (1/2)b(hb + 3hs)
Surface area of a right cone: πr(r + h)
Surface area of a sphere: 4πr^2
Volume of a square pyramid: (1/3)b^2 × H
Volume of a triangular pyramid: (1/3) × (1/2)bh × H
Volume of a right cone: (1/3)πr^2 × H
Volume of a sphere: (4/3)πr^3

Effects of Scaling

If one or more dimensions of a prism or cylinder are multiplied by a constant factor k, the surface area and volume will change
Volume scales by k^3
Surface area scales by k^2

Circle Geometry

Definitions:
- Arc: a portion of the circumference of a circle
- Chord: a straight line joining two points on a circle
- Circumference: the perimeter or boundary line of a circle
- Radius: a line from the center of a circle to a point on the circumference
- Diameter: a special chord that passes through the center of the circle
- Segment: a part of the circle cut off by a chord
- Tangent: a straight line that touches the circle at only one point
Theorem of Pythagoras: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides
Theorems:
- A tangent to a circle is perpendicular to the radius drawn to the point of contact
- Perpendicular line from circle center bisects chord
- Perpendicular bisector of chord passes through circle center
- Angle at the center is twice the angle at the circumference
- Angles subtended by the same arc are equal
- Opposite angles of a cyclic quadrilateral are supplementary
- Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle
- Tangent-chord theorem: the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle which the chord subtends in the alternate segment

The Sine Function

Period: 360°
Amplitude: 1
Domain: [0°, 360°]
Range: [-1, 1]
x-intercepts: (0°, 0), (180°, 0), (360°, 0)
y-intercept: (0°, 0)
Maximum turning point: (90°, 1)
Minimum turning point: (270°, -1)
Effects of a and q on the graph of y = a sin(θ) + q
Effects of k on the graph of y = sin(kθ)

The Cosine Function

Period: 360°
Amplitude: 1
Domain: [0°, 360°]
Range: [-1, 1]
x-intercepts: (90°, 0), (270°, 0)
y-intercept: (0°, 1)
Maximum turning points: (0°, 1), (360°, 1)
Minimum turning point: (180°, -1)
Effects of a and q on the graph of y = a cos(θ) + q
Effects of k on the graph of y = cos(kθ)

The Tangent Function

Period: 180°
Domain: {θ: 0° ≤ θ ≤ 360°, θ ≠ 90°, 270°}
Range: {f(θ): f(θ) ∈ ℝ}
x-intercepts: (0°, 0), (180°, 0), (360°, 0)
y-intercept: (0°, 0)
Asymptotes: θ = 90°, θ = 270°
Effects of a and q on the graph of y = a tan(θ) + q
Effects of k on the graph of y = tan(kθ)

Here are the study notes for the provided text:

Equation of a Line*

Two-Point Form

The equation of a straight line can be derived from two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2)
Formula: y−y1x−x1=y2−y1x2−x1\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}x−x1y−y1=x2−x1y2−y1

Gradient-Point Form

Derived from the definition of gradient and the two-point form
Formula: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1=m(x−x1)
Requires the gradient of the line and the coordinates of one point on the line

Gradient-Intercept Form

Derived from the gradient-point form
Formula: y=mx+cy = mx + cy=mx+c
ccc is the y-intercept of the straight line
mmm is the gradient of the line
Inclination of a Line*

Relationship between Gradient and Inclination

Gradient mmm of a line is equal to the tangent of the angle θ\thetaθ it makes with the positive x-axis
Formula: m=tan⁡θm = \tan \thetam=tanθ for 0∘≤θ<180∘0^\circ \leq \theta < 180^\circ0∘≤θ<180∘

Special Cases

Vertical lines: θ=90∘\theta = 90^\circθ=90∘, mmm is undefined
Horizontal lines: θ=0∘\theta = 0^\circθ=0∘, m=0m = 0m=0
Lines with negative gradients: m<0m < 0m<0, tan⁡θ<0\tan \theta < 0tanθ<0
Parallel Lines*

Gradient Relationship

Two lines are parallel if and only if their gradients are equal
Formula: m1=m2m_1 = m_2m1=m2

Finding the Equation of a Parallel Line

Identify the gradient of the given line
Use the point-slope form of the equation of a line
Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
Perpendicular Lines*

Gradient Relationship

Two lines are perpendicular if and only if the product of their gradients is equal to -1
Formula: m1×m2=−1m_1 \times m_2 = -1m1×m2=−1

Finding the Equation of a Perpendicular Line

Identify the gradient of the given line
Calculate the perpendicular gradient
Use the point-slope form of the equation of a line
Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
Trigonometric Identities*

Quotient Identity

Formula: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ
Defined for all values of θ\thetaθ except where cos⁡θ=0\cos \theta = 0cosθ=0

Square Identity

Formula: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
Alternative forms: sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \thetasin2θ=1−cos2θ, cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \thetacos2θ=1−sin2θ
Reduction Formulae*

Angle Sum and Difference Formulae

Formulae for sine, cosine, and tangent of 90∘±θ90^\circ \pm \theta90∘±θ, 180∘±θ180^\circ \pm \theta180∘±θ, and 360∘±θ360^\circ \pm \theta360∘±θ
Reduction formulae allow for the expression of trigonometric functions in terms of acute angles, facilitating simplification and evaluation
Trigonometric Equations*

Solving Trigonometric Equations

Use a calculator to find the reference angle
Use the CAST diagram to determine the quadrants
Use reduction formulae to find the values of the angle
Check the solution using a calculator
Area of a Polygon*

Types of Polygons

Square: Area=s2Area = s^2Area=s2
Rectangle: Area=b×hArea = b \times hArea=b×h
Triangle: Area=12b×hArea = \frac{1}{2} b \times hArea=21b×h
Trapezium: Area=12(a+b)×hArea = \frac{1}{2} (a + b) \times hArea=21(a+b)×h
Parallelogram: Area=b×hArea = b \times hArea=b×h
Circle: Area=πr2Area = \pi r^2Area=πr2, Circumference = 2πr2\pi r2πr
Right Prisms and Cylinders*

Surface Area

Calculate the area of each face and add them together
Formulas for surface area of rectangular prism, cube, triangular prism, and cylinder

Volume

Formula: Volume=Area×HeightVolume = Area \times HeightVolume=Area×Height
Formulas for volume of rectangular prism, cube, triangular prism, and cylinder### Rectangular Prism, Triangular Prism, and Cylinder
Volume of a rectangular prism: l × b × h
Volume of a triangular prism: (1/2)b × h × H
Volume of a cylinder: πr^2 × h

Right Pyramids, Right Cones, and Spheres

Definition of a pyramid: a geometric solid with a polygon base and sides that converge at a point (apex)
Definition of a right pyramid: a pyramid with a line from the apex to the center of the base perpendicular to the base
Definition of a cone: a geometric solid with a circular base and sides that converge at a point (apex)
Definition of a sphere: a perfectly round solid, looking the same from any direction

Surface Area and Volume of Pyramids, Cones, and Spheres

Surface area of a square pyramid: b(b + 2hs)
Surface area of a triangular pyramid: (1/2)b(hb + 3hs)
Surface area of a right cone: πr(r + h)
Surface area of a sphere: 4πr^2
Volume of a square pyramid: (1/3)b^2 × H
Volume of a triangular pyramid: (1/3) × (1/2)bh × H
Volume of a right cone: (1/3)πr^2 × H
Volume of a sphere: (4/3)πr^3

Effects of Scaling

If one or more dimensions of a prism or cylinder are multiplied by a constant factor k, the surface area and volume will change
Volume scales by k^3
Surface area scales by k^2

Circle Geometry

Definitions:
- Arc: a portion of the circumference of a circle
- Chord: a straight line joining two points on a circle
- Circumference: the perimeter or boundary line of a circle
- Radius: a line from the center of a circle to a point on the circumference
- Diameter: a special chord that passes through the center of the circle
- Segment: a part of the circle cut off by a chord
- Tangent: a straight line that touches the circle at only one point
Theorem of Pythagoras: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides
Theorems:
- A tangent to a circle is perpendicular to the radius drawn to the point of contact
- Perpendicular line from circle center bisects chord
- Perpendicular bisector of chord passes through circle center
- Angle at the center is twice the angle at the circumference
- Angles subtended by the same arc are equal
- Opposite angles of a cyclic quadrilateral are supplementary
- Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle
- Tangent-chord theorem: the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle which the chord subtends in the alternate segment

The Sine Function

Period: 360°
Amplitude: 1
Domain: [0°, 360°]
Range: [-1, 1]
x-intercepts: (0°, 0), (180°, 0), (360°, 0)
y-intercept: (0°, 0)
Maximum turning point: (90°, 1)
Minimum turning point: (270°, -1)
Effects of a and q on the graph of y = a sin(θ) + q
Effects of k on the graph of y = sin(kθ)

The Cosine Function

Period: 360°
Amplitude: 1
Domain: [0°, 360°]
Range: [-1, 1]
x-intercepts: (90°, 0), (270°, 0)
y-intercept: (0°, 1)
Maximum turning points: (0°, 1), (360°, 1)
Minimum turning point: (180°, -1)
Effects of a and q on the graph of y = a cos(θ) + q
Effects of k on the graph of y = cos(kθ)

The Tangent Function

Period: 180°
Domain: {θ: 0° ≤ θ ≤ 360°, θ ≠ 90°, 270°}
Range: {f(θ): f(θ) ∈ ℝ}
x-intercepts: (0°, 0), (180°, 0), (360°, 0)
y-intercept: (0°, 0)
Asymptotes: θ = 90°, θ = 270°
Effects of a and q on the graph of y = a tan(θ) + q
Effects of k on the graph of y = tan(kθ)

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Description

Derive and use different forms of the straight line equation, including two-point, gradient-point, and gradient-intercept forms, depending on the problem information.