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?
What is the value of y when x is 0 in the gradient–intercept form of a straight line equation?
What is the definition of the constant c in the gradient–intercept form of a straight line equation?
What form of the straight line equation is derived from the two-point form?
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What are the two pieces of information required to use the gradient–point form of a straight line equation?
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What is the two-point form of the straight line equation used for?
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What is the step to derive the gradient–intercept form from the gradient–point form?
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What is the name of the form of the straight line equation that is used when two points on the line are known?
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What is the relationship between the gradient of a line and its inclination?
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What is the gradient of a vertical line?
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If two lines are parallel, what can be said about their gradients?
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What is the formula to find the inclination of a line with a negative gradient?
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What is the purpose of the point-slope form of a line?
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Why is it important to ensure the equation of a line is in gradient-intercept form?
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What can be said about the tangent of an acute angle?
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What is the gradient of a horizontal line?
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What is the relationship between the inclination of a line and the positive x-axis?
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What is the purpose of using the CAST diagram in trigonometry?
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What is the condition for two lines to be parallel?
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If two lines have gradients of 2 and -1/2, what can be concluded about the lines?
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What is the equation of a line perpendicular to y = 3x + 2 and passes through the point (1, 4)?
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What is the trigonometric identity for tangent in terms of sine and cosine?
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What is the Pythagorean identity in trigonometry?
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What is a useful tip for proving trigonometric identities?
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What does the square identity simplify to?
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What is a condition for tan θ to be undefined?
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What is the first step in finding the equation of a line perpendicular to a given line?
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What is the point-slope form of the equation of a line?
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What is the value of tan(θ) in terms of sine and cosine functions?
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What is the result of the expression sin²(θ) + cos²(θ)?
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What is the value of sin(180° - θ) in terms of sine of θ?
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What is the value of cos(180° + θ) in terms of cosine of θ?
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What is the value of tan(180° + θ) in terms of tangent of θ?
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What is the expression for sin²(θ) in terms of cosine of θ?
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What is the expression for cos²(θ) in terms of sine of θ?
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What is the characteristic of the reduction formula for trigonometric functions?
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If point P is rotated through 90°, what are the coordinates of P'?
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What is the value of sin(180° + θ)?
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If points P and P' are symmetrical about the x-axis, what are the coordinates of P'?
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What is the value of cos(360° - θ)?
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What is the value of tan(180° + θ)?
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If points P and P' are symmetrical about the origin, what are the coordinates of P'?
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What is the value of sin(-θ)?
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What is the value of cos(90° + θ)?
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What is the value of tan(90° - θ)?
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What is the value of sin(360° + θ)?
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What is the formula for the surface area of a square pyramid?
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What is the formula for the volume of a triangular prism?
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If the dimensions of a solid are multiplied by a factor k, how does the volume scale?
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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?
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What is the formula for the area of a circle?
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What is the name of the line that touches a circle at only one point?
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What is the formula for the surface area of a right cylinder?
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What is the formula for the volume of a sphere?
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What is the name of the theorem that states that the perpendicular bisector of a chord passes through the center of the circle?
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What is the formula for the area of a trapezium?
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A line segment from the center of a circle is perpendicular to a chord. What is the effect on the chord?
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What is the relationship between the angles subtended by the same arc in a circle?
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What is the period of the function y = sin(kθ)?
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What is the effect of a on the shape of the function y = a sin(θ)?
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What is the effect of q on the graph of the function y = a sin(θ) + q?
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What are the x-intercepts of the function y = cos(θ)?
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What is the effect of k on the period of the function y = sin(kθ)?
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What is the range of the function y = sin(θ)?
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What is the effect of a on the amplitude of the function y = a cos(θ)?
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What is the y-intercept of the function y = cos(θ)?
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What is the primary purpose of using reduction formulae and co-function rules in trigonometry?
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What is the first step in solving a trigonometric equation?
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What is the formula for the area of a trapezium?
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What is the surface area of a prism or cylinder?
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What is the name of the geometric solid that has a polygon as its base and vertical sides perpendicular to the base?
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What is the primary purpose of unfolding a prism or cylinder into a net?
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What is the formula for the area of a circle?
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What is the general approach to solving quadratic trigonometric equations?
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What is the formula for the circumference of a circle?
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What is the purpose of using the CAST diagram in trigonometry?
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What is the formula to calculate the volume of a triangular prism?
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What is the formula to calculate the surface area of a right cone?
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What is the formula to calculate the volume of a cylinder?
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If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new volume?
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What is the formula to calculate the surface area of a sphere?
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What is the formula to calculate the volume of a right pyramid?
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If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new surface area?
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What is the formula to calculate the volume of a square pyramid?
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What is the formula to calculate the surface area of a triangular pyramid?
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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?
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What is the period of the function y = cos(2θ)?
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What is the effect of k on the period of y = tan(kθ)?
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What is the range of y = tan(θ)?
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What is the effect of p on the graph of y = tan(θ + p)?
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What is the domain of y = cos(kθ)?
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What is the effect of a on the graph of y = a tan(θ)?
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What is the y-intercept of y = cos(kθ)?
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What is the effect of k on the graph of y = cos(kθ)?
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What is the domain of y = tan(kθ)?
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What is the asymptote of y = tan(kθ)?
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What is the value of sin(180° - θ) in terms of sine of θ?
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What is the condition for tan(θ) to be undefined?
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What is the expression for cos²(θ) in terms of sine of θ?
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What is the value of tan(180° + θ) in terms of tangent of θ?
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What is the characteristic of the reduction formula for trigonometric functions?
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What is the value of cos(360° - θ) in terms of cosine of θ?
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What is the proof of the quotient identity tan(θ) = sin(θ) / cos(θ)?
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What is the result of the expression sin²(θ) + cos²(θ)?
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What is the main difference between the two-point form and the gradient-point form of the straight line equation?
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What is the purpose of rearranging the gradient-point form to make y the subject of the formula?
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What is the relationship between the gradient of a line and its inclination?
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What can be said about the y-intercept of a line in the gradient-intercept form?
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What is the advantage of using the gradient-intercept form of a straight line equation?
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What is the condition for two lines to be parallel?
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What is the purpose of using the two-point form of a straight line equation?
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What can be said about the gradient of a vertical line?
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What is the relationship between the gradient of a line and its inclination in the second quadrant?
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What is the equation of a line that passes through the point (2, 3) and is parallel to the line y = 2x + 1?
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What is the value of θ when the gradient of a line is -3/4?
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What is the condition for two lines to be parallel in terms of their inclination?
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What is the equation of a line that passes through the point (1, 2) and has a gradient of 3/2?
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What is the gradient of a line that is perpendicular to the line y = 2x + 1 and passes through the origin?
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What is the value of θ when the gradient of a line is 2/3?
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What is the equation of a line that is parallel to the line y = 3x - 2 and passes through the point (4, 5)?
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What is the relationship between the gradient of a line and its inclination in the first quadrant?
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What is the equation of a line that passes through the point (3, 4) and is parallel to the line y = 2x + 3?
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If two lines are perpendicular, what is the relationship between their gradients?
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What is the purpose of the point-slope form of a line?
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What is the trigonometric identity for sin²(θ) + cos²(θ)
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What is the condition for tan(θ) to be undefined?
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If two lines have gradients of 3 and -1/3, what can be concluded about the lines?
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What is the first step in finding the equation of a line perpendicular to a given line?
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What is the result of the expression sin(θ)cos(θ) + cos(θ)sin(θ)?
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What is the value of tan(θ) in terms of sine and cosine functions?
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What is the useful tip for proving trigonometric identities?
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What is the equation of a line perpendicular to y = 2x - 3 and passing through the point (1, 5)?
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If points P and P' are symmetrical about the line y = x, what are the coordinates of P'?
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What is the value of sin(360° - θ) in terms of sine of θ?
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If P is rotated through 90°, what are the coordinates of P'?
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What is the value of cos(180° + θ) in terms of cosine of θ?
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What is the value of tan(90° + θ) in terms of tangent of θ?
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What is the value of sin(90° - θ) in terms of cosine of θ?
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If points P and P' are symmetrical about the origin, what are the coordinates of P'?
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What is the value of cos(360° + θ) in terms of cosine of θ?
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What is the value of tan(180° - θ) in terms of tangent of θ?
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What is the value of sin(-θ) in terms of sine of θ?
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What is the purpose of using the CAST diagram in trigonometry?
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What is the reduction formula for tan(180° + θ)?
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What is the general solution of the equation sin(θ) = 1/2?
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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?
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What is the formula for the area of a trapezium?
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What is the value of sin(90° + θ) in terms of cosine of θ?
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What is the characteristic of the reduction formula for trigonometric functions?
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What is the formula for the area of a circle?
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What is the purpose of unfolding a prism into a net?
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What is the value of cos(180° - θ) in terms of cosine of θ?
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What is the formula for the volume of a cylinder?
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What is the formula for the surface area of a sphere?
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What is the effect of scaling the dimensions of a solid by a factor k on its volume?
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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?
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What is the formula for the surface area of a triangular pyramid?
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What is the formula for the volume of a square pyramid?
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What is the effect of a on the shape of the function y = a sin θ?
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What is the formula for the area of a trapezium?
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What is the period of the function y = sin kθ?
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What is the formula for the area of a circle?
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What is the formula for the volume of a triangular prism?
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What is the effect of p on the graph of the function y = sin(θ + p)?
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What is the y-intercept of the function y = cos θ?
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What is the effect of scaling the dimensions of a solid by a factor k on its surface area?
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What is the amplitude of the function y = a sin θ + q?
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What is the effect of q on the graph of the function y = a sin θ + q?
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What is the formula for the volume of a right cone?
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What is the x-intercept of the function y = sin θ?
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What is the formula for the surface area of a sphere?
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What is the range of the function y = sin kθ?
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What is the domain of the function y = cos θ?
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If the dimensions of a solid are multiplied by a factor k, how does the surface area scale?
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What is the minimum turning point of the function y = cos θ?
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What is the formula for the volume of a triangular pyramid?
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What is the formula for the surface area of a right cone?
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If the dimensions of a solid are multiplied by a factor k, how does the volume scale?
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What is the formula for the volume of a rectangular prism?
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What is the formula for the surface area of a square pyramid?
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What is the formula for the volume of a cylinder?
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What is the formula for the surface area of a triangular prism?
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What is the period of the function y = cos kθ?
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What is the effect of k on the graph of y = cos kθ when k > 1?
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What is the y-intercept of the function y = cos kθ?
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What is the effect of p on the graph of y = cos(θ + p) when p > 0?
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What is the period of the function y = tan kθ?
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What is the effect of k on the graph of y = tan kθ when k > 1?
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What is the y-intercept of the function y = tan kθ?
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What is the effect of p on the graph of y = tan(θ + p) when p > 0?
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What is the range of the function y = tan kθ?
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What is the domain of the function y = tan kθ?
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What is the condition required to use the two-point form of the straight line equation?
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What is the formula derived from the gradient–point form of the straight line equation?
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What is the value of y_1 - mx_1 in the gradient–intercept form of the straight line equation?
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What is the purpose of the gradient–point form of the straight line equation?
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What is the relationship between the gradient of a line and the constant c in the gradient–intercept form?
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What is the advantage of using the gradient–intercept form of the straight line equation?
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What is the formula derived from the two-point form of the straight line equation?
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What is the condition required to use the gradient–intercept form of the straight line equation?
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What is the condition for tan θ to be undefined?
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What is the result of the expression sin²(θ) + cos²(θ)?
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What is the value of sin(180° - θ) in terms of sine of θ?
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What is the expression for sin²(θ) in terms of cosine of θ?
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What is the characteristic of the reduction formula for trigonometric functions?
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What is the value of tan(180° + θ) in terms of tangent of θ?
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What is the relationship between the gradient of a line and its inclination when the angle is acute?
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What is the value of cos(180° - θ) in terms of cosine of θ?
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What is the value of sin(90° + θ) in terms of cosine of θ?
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Which form of the equation of a line is most useful when finding the equation of a parallel line?
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If a line has a negative gradient, what can be said about the angle of inclination?
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What is the relationship between the gradient of a line and its y-intercept in the gradient-intercept form?
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What is the purpose of using the CAST diagram in trigonometry when dealing with negative gradients?
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What is the condition for two lines to be parallel in terms of their gradients?
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What is the step to derive the equation of a parallel line from a given line?
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What is the trigonometric relationship between the gradient of a line and its inclination when the angle is obtuse?
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What is the importance of ensuring the equation of a line is in gradient-intercept form when finding the equation of a parallel line?
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What is the characteristic of the tangent function in the second quadrant?
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What is the condition for two lines to be perpendicular in terms of their gradients?
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What is the trigonometric identity for sine in terms of cosine?
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What is the first step in finding the equation of a line perpendicular to a given line?
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What is the purpose of the square identity in trigonometry?
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What is the condition for tanθ to be undefined?
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What is the result of the expression sin²(θ) + cos²(θ)?
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What is the gradient of a line perpendicular to y = 2x + 3?
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What is the equation of a line perpendicular to y = 3x + 2 and passes through the point (2, 5)?
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What is the trigonometric identity for cosine in terms of sine?
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What is the purpose of using the quotient identity in trigonometry?
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What is the formula for the surface area of a right cone?
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What is the formula for the volume of a cylinder?
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If the dimensions of a solid are multiplied by a factor k, how does the surface area scale?
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What is the formula for the surface area of a right cone?
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What is the formula for the volume of a sphere?
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What is the formula for the area of a trapezium?
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Which of the following trigonometric identities is used to express trigonometric functions in terms of acute angles?
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If the dimensions of a rectangular prism are multiplied by a factor k, how does the surface area change?
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What is the first step in solving a trigonometric equation?
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What is the formula for the volume of a sphere?
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What is the formula for the area of a circle?
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If the length of a rectangle is multiplied by a factor k, how does the volume of the rectangular prism change?
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What is the theorem that states that the angle at the center of a circle is twice the angle at the circumference?
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What is the formula for the area of a trapezium?
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What is the formula for the surface area of a triangular pyramid?
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What is the name of the geometric solid that has a polygon as its base and vertical sides perpendicular to the base?
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What is the formula for the area of a parallelogram?
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What is the purpose of unfolding a prism or cylinder into a net?
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If the height of a cylinder is multiplied by a factor k, how does the volume change?
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What is the formula for the volume of a square pyramid?
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What is the formula for the volume of a triangular prism?
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What is the formula for the area of a triangle?
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What is the formula for the area of a circle?
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If the radius of a sphere is multiplied by a factor k, how does the volume change?
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What is the characteristic of the reduction formula for trigonometric functions?
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What is the formula for the surface area of a square pyramid?
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What is the purpose of using the CAST diagram in trigonometry?
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If the base of a right pyramid is multiplied by a factor k, how does the surface area change?
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What is the formula for the circumference of a circle?
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What is the formula for the area of a triangle?
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What is the effect of a > 1 on the graph of y = a sin θ?
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What is the period of the graph of y = sin kθ?
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What is the minimum turning point of the graph of y = cos θ?
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What is the effect of q on the graph of y = a sin θ + q?
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What is the domain of the function y = sin kθ?
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What is the range of the function y = a sin θ?
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What is the effect of p on the graph of y = sin(θ + p)?
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What is the x-intercept of the graph of y = sin θ?
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What is the maximum turning point of the graph of y = cos θ?
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What is the y-intercept of the graph of y = a cos θ + q?
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If points P and P' are symmetrical about the y-axis, what are the coordinates of P'?
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What is the value of sin(360° - θ) in terms of sine of θ?
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If points P and P' are symmetrical about the line y = x, what are the coordinates of P'?
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What is the value of cos(90° + θ) in terms of sine of θ?
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What is the value of tan(90° - θ) in terms of tangent of θ?
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What is the value of sin(180° + θ) in terms of sine of θ?
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What is the value of cos(360° + θ) in terms of cosine of θ?
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What is the value of tan(180° - θ) in terms of tangent of θ?
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What is the value of sin(-θ) in terms of sine of θ?
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What is the value of cos(90° - θ) in terms of cosine of θ?
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What is the period of the function y = cos kθ?
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What is the effect of k on the period of y = cos kθ?
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What is the range of the function y = cos kθ?
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What is the effect of p on the graph of y = cos(θ + p)?
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What is the period of the function y = tan kθ?
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What is the effect of k on the period of y = tan kθ?
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What is the domain of the function y = tan kθ?
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What is the effect of p on the graph of y = tan(θ + p)?
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What is the effect of a on the shape of the graph of y = a tan θ + q?
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What is the range of the function y = tan kθ?
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Which of the following is true about the gradient–point form of a straight line equation?
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What is the advantage of expressing a straight line equation in the gradient–intercept form?
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Given a straight line equation in the form y = mx + c, what can be said about the value of c?
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Which of the following is a step in deriving the gradient–intercept form from the gradient–point form?
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What is the significance of the two-point form of a straight line equation?
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Which of the following is a characteristic of the gradient–point form of a straight line equation?
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What is the relationship between the two-point form and the gradient–point form of a straight line equation?
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What is the advantage of expressing a straight line equation in the two-point form?
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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=tanθm = \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, tanθ<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: tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ
- Defined for all values of θ\thetaθ except where cosθ=0\cos \theta = 0cosθ=0
Square Identity
- Formula: sin2θ+cos2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
- Alternative forms: sin2θ=1−cos2θ\sin^2 \theta = 1 - \cos^2 \thetasin2θ=1−cos2θ, cos2θ=1−sin2θ\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=tanθm = \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, tanθ<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: tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ
- Defined for all values of θ\thetaθ except where cosθ=0\cos \theta = 0cosθ=0
Square Identity
- Formula: sin2θ+cos2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
- Alternative forms: sin2θ=1−cos2θ\sin^2 \theta = 1 - \cos^2 \thetasin2θ=1−cos2θ, cos2θ=1−sin2θ\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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Derive and use different forms of the straight line equation, including two-point, gradient-point, and gradient-intercept forms, depending on the problem information.