Gr 12 Mathematics: November Hard P(2)

What is the equation of a circle with center at (a, b) and radius r?

What is the purpose of completing the square in the equation of a circle?

What is the relationship between the radius and the tangent line at the point of tangency?

What is the key concept that ensures the tangent line touches the circle at only one point?

What is the formula for the center of a circle given the general form equation x^2 + y^2 + Dx + Ey + F = 0?

What is the formula for the radius of a circle given the general form equation x^2 + y^2 + Dx + Ey + F = 0?

What is the definition of a tangent line to a circle?

What is the purpose of the equation of a circle?

What is the relationship between the gradient of the radius and the gradient of the tangent?

What is the next step after finding the center of the circle in determining the equation of the tangent?

What is the property of proportion that states wz = xy?

What is the purpose of the Basic Proportionality Theorem in geometry?

What is the formula for the area of a triangle?

What is the definition of a polygon?

What is the formula for the gradient of the tangent?

What is the first step in solving proportional problems?

What is the name of the theorem that states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

What is the purpose of ratios in geometry?

What is the relationship between the sides of two triangles if the corresponding angles are equal?

In a triangle, if a line drawn from the midpoint of one side is parallel to another side, what conclusion can be drawn about the third side?

What must be true for two polygons to be classified as similar?

When two triangles have their corresponding sides in proportion, what can we conclude about their angles?

What is the condition for two triangles to be similar?

What does the Proportion Theorem state about a line drawn parallel to one side of a triangle?

In the context of similarity, what is required to prove that two triangles are equiangular?

What is the purpose of the Pythagorean theorem?

What is the equation of the linear regression line?

If two polygons are similar, what can you say about their areas?

What is the range of the regression coefficient (r)?

Which condition is NOT required for two triangles to be similar?

What is the formula for the area of a triangle?

In a triangle, if side lengths are $6$, $8$, and $10$, what can you conclude about the triangle?

What can be concluded about the lengths of sides DE and BC if DE is drawn parallel to BC and DE is half the length of BC?

What is the converse of the Pythagorean theorem?

What is the purpose of the linear regression line?

What is the condition for two triangles to be congruent?

What is the formula for the Pythagorean theorem?

What does a regression coefficient (r) close to -1 indicate?

What is the area of a rhombus if the lengths of the diagonals AC and BD are 8 units and 6 units respectively?

If two triangles are equiangular, what can be inferred about their sides?

According to the Triangle Proportionality Theorem, what does a line drawn parallel to one side of a triangle do to the other two sides?

What is the height of a trapezium if the bases are of lengths 10 units and 14 units and the area is 120 square units?

In a triangle with a base of 12 units and height of 3 units, what is the area?

What is the relationship between the sides of two similar polygons?

If the diagonal AC of a kite measures 10 units and diagonal BD measures 8 units, what is the area of the kite?

In a right triangle, if one leg measures 6 units and the other leg measures 8 units, what is the length of the hypotenuse?

If a line segment connects the midpoints of two sides of a triangle, what is the length of that segment compared to the third side?

What must be true for two triangles to be considered equal in area but not necessarily congruent?

What does the distance formula express when comparing two points on the unit circle?

Which identity is used to derive the cosine of the sum of two angles?

From the cosine rule, what does the expression $KL^2 = 2 - 2 ext{cos}( heta)$ imply for the relationship of points on the unit circle?

After applying the even-odd identities, what is the final expression for $cos(\alpha + \beta)$?

When using the cosine rule, what is the relationship between $\cos(\alpha - \beta)$ and the cosine and sine components of $\alpha$ and $\beta$?

What type of relationship is established by the equation $2 - 2 \cos(\alpha - \beta) = 2 - 2 (\cos \alpha \cos \beta + \sin \alpha \sin \beta)$?

What is the interpretation of a correlation coefficient of 0.8?

What is the formula for calculating the linear correlation coefficient r?

What is the equation of a circle with center at the origin and radius r?

What is the cosine of a difference formula in trigonometry?

What is the interpretation of a correlation coefficient of -0.6?

What is the sine of a sum formula in trigonometry?

What is the condition for a circle to be symmetric about the x-axis?

What is the formula for calculating the y-intercept of a linear regression line?

What is the interpretation of a correlation coefficient of 0.2?

What is the cosine of a sum formula in trigonometry?

What is the formula for the sine of a difference of two angles?

What is the formula for the cosine of a sum of two angles?

What is the formula for the sine of a double angle?

What is the formula for the cosine of a double angle?

What is the general method for solving trigonometric equations?

What is the purpose of using the CAST diagram in solving trigonometric equations?

What is the formula for the sine of a sum of two angles?

What is the formula for the cosine of a difference of two angles?

What is the purpose of finding the general solution in solving trigonometric equations?

What is the formula for the cosine of a double angle in terms of sine?

What is the general solution for $ an \theta = x$?

Which of the following scenarios would require the Cosine Rule?

In calculating the area of triangle ABC using the Area Rule, which formula is valid?

To find the height of a building from point B using triangle BCD, what does BD in the equation represent?

Which trigonometric function is used to relate FB and the angle $eta$ in the height of a pole problem?

In three-dimensional problems, which initial step is crucial for solving the problem?

When is the Sine Rule applicable in triangle calculations?

For the equation $b^2 = a^2 + c^2 - 2ac \cos B$, what does the term $-2ac \cos B$ represent?

In a triangle ABC, point D lies on side AB and point E lies on side AC such that DE is parallel to BC. If AD = 4, DB = 6, and AE = 5, what is the length of EC?

Triangle PQR and triangle SQR share the same base QR. If the area of triangle PQR is 12 square units and the area of triangle SQR is 8 square units, what can you conclude about the relationship between PQ and SR?

A line segment connects the midpoints of two sides of a triangle. If the length of the third side of the triangle is 10 units, what is the length of the line segment connecting the midpoints?

Two triangles are similar. The ratio of the corresponding sides of the two triangles is 3:5. If the area of the smaller triangle is 18 square units, what is the area of the larger triangle?

A parallelogram has a base of length 12 units and a height of 8 units. What is the area of the parallelogram?

A kite has diagonals of length 10 units and 6 units. What is the area of the kite?

A trapezium has bases of length 10 units and 16 units, and a height of 8 units. What is the area of the trapezium?

In a right-angled triangle, the lengths of the two shorter sides are 5 units and 12 units. What is the length of the hypotenuse?

A triangle has a base of length 10 units and a height of 6 units. What is the area of the triangle?

A rhombus has diagonals of length 8 units and 6 units. What is the area of the rhombus?

Given a circle with the equation (x^2 + y^2 - 6x + 4y - 12 = 0), what is the equation of the tangent line to the circle at the point (8, 2)?

A circle has a center at ( (-2, 3) ) and passes through the point ( (1, 5) ). What is the equation of the circle?

A circle has the equation (x^2 + y^2 + 8x - 6y - 11 = 0). What is the radius of the circle?

Find the equation of the tangent line to the circle (x^2 + y^2 - 4x + 6y - 3 = 0) at the point (5, 1).

Given the circle (x^2 + y^2 - 10x + 4y - 20 = 0), what is the equation of the tangent line at the point (7, 3)?

A circle has the equation (x^2 + y^2 - 12x + 8y + 16 = 0). What is the center of the circle?

A circle has a diameter with endpoints ( (-3, 5) ) and ( (1, 1) ). What is the equation of the circle?

Given a circle with the equation (x^2 + y^2 + 4x - 10y + 20 = 0), what is the equation of the tangent line at the point (2, 4)?

In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If DE is parallel to BC, which of the following statements is true?

Using the distance formula and cosine rule, we can derive the angle difference formula. What key trigonometric identity is used in the derivation of the angle sum formula, ( \cos(\alpha + \beta) )?

What step in the derivation of ( \cos(\alpha + \beta) ) directly utilizes the relationship established through the distance formula and cosine rule?

Which of the following statements accurately describes the relationship between the distance formula, the cosine rule, and the angle difference formula?

The derivation of the angle sum formula for cosine demonstrates the power of using trigonometric identities. Which of the following identities is NOT directly utilized in the derivation?

The derivation of ( \cos(\alpha + \beta) ) hinges on the ability to express the angle sum as a difference. How does the negative angle identity for cosine facilitate this transformation?

What value of the correlation coefficient indicates a strong positive correlation?

What does it mean when the correlation coefficient, $r$, is equal to $-0.7$?

How is the linear correlation coefficient $r$ calculated using the regression coefficient $b$?

What is the equation of a circle with a center at the origin and a radius of 4?

Which of the following best describes a scenario where $r = 0$?

In the context of correlation, what is true when $r$ is very close to -1?

How is the regression line defined mathematically?

What are the limits that define a weak positive correlation?

What is the significance of the circle equation $x^2 + y^2 = r^2$?

What is the correct expansion of (\cos(3\alpha)) in terms of (\cos\alpha) and (\sin\alpha)?

Given (\sin(2\theta) = \frac{2}{5}), what is the value of (\cos(4\theta))?

If (\cos(x) = \frac{3}{5}) and (0 < x < \frac{\pi}{2}), then what is the value of (\sin(2x))?

Given that (\cos(A + B) = \frac{1}{2}) and (\sin(A) = \frac{3}{5}), where (0 < A < \frac{\pi}{2}) and (0 < B < \frac{\pi}{2}), find the value of (\cos(B)).

Find the general solution to the equation (2\sin^2(x) - 3\sin(x) + 1 = 0) for (0 \leq x < 2\pi).

If (\sin(x) = \frac{1}{3}) and (\cos(y) = \frac{2}{3}), where (0 < x < \frac{\pi}{2}) and (0 < y < \frac{\pi}{2}), what is the value of (\cos(x - y))?

Simplify the expression: (\frac{\sin(2x)}{1 + \cos(2x)})

What is the value of (\cos(15^\circ))?

If (\sin(x) + \cos(x) = \frac{\sqrt{2}}{2}), find the value of (\sin(2x)).

Prove the identity: (\frac{1 - \cos(2x)}{\sin(2x)} = \tan(x))

What is the correct general solution for the equation \( an heta = x \)?

When should the Sine Rule be applied in triangle problems?

Which equation is used to calculate the area of triangle ABC when two sides and the included angle are known?

What is the correct formula to find the height of a pole given in triangle FAB?

Which of the following statements correctly describes the use of the Cosine Rule?

What condition must be met to apply the Area Rule for a triangle?

For which circumstance should the Sine Rule NOT be applied?

In three-dimensional problems, which step is crucial before applying trigonometric rules?

A circle has the equation ( (x - 3)^2 + (y + 2)^2 = 25 ). A tangent line to the circle passes through the point (8, 1). What is the slope of the tangent line?

In a triangle ( ABC ), a line ( DE ) is drawn parallel to side ( BC ) such that ( AD = 4 ), ( DB = 6 ), and ( AE = 5 ). What is the length of ( EC )?

If the radius of a circle is 5 units and a tangent line to the circle passes through the point (12, 5), what is the possible equation of the tangent line?

In a triangle ( ABC ), line ( DE ) is parallel to side ( BC ), and ( AD/DB = 3/4 ). If the area of triangle ( ADE ) is 12 square units, what is the area of triangle ( ABC )?

A circle has the equation ( (x + 2)^2 + (y - 1)^2 = 16 ). A tangent line to the circle is perpendicular to the line ( y = 2x + 5 ). What is the equation of this tangent line?

In a triangle ( ABC ), line ( DE ) is parallel to side ( BC ), with ( D ) on ( AB ) and ( E ) on ( AC ). If ( AD = 3 ), ( DB = 5 ), and ( BC = 12 ), what is the length of ( DE )?

Given a circle with the equation ( (x - 4)^2 + (y + 3)^2 = 9 ) and a tangent line passing through the point (7, 1), what is the equation of the tangent line?

In a triangle ( ABC ), line ( DE ) is parallel to side ( BC ), with ( D ) on ( AB ) and ( E ) on ( AC ). If ( AD/DB = 2/3 ) and ( BC = 10 ), what is the length of ( DE )?

A circle has a diameter with endpoints at (2, 5) and (8, -1). What is the equation of the tangent line to this circle at the point (8, -1)?

Two triangles, ( ABC ) and ( DEF ), are similar, with ( AB = 6 ), ( BC = 8 ), and ( DE = 9 ). What is the perimeter of triangle ( DEF )?

In the proof of the Pythagorean theorem, why is \(\triangle ABD\) similar to \(\triangle CBA\)?

In a triangle ABC, a line DE is drawn parallel to BC such that D lies on AB and E lies on AC. If AD = 4 units, DB = 6 units, and AE = 5 units, what is the length of EC?

In two similar triangles, if the ratio of the lengths of their corresponding sides is 2:3, what is the ratio of their areas?

If two triangles are equiangular, what can be concluded about their corresponding sides?

What is the converse of the Mid-point Theorem?

What is the condition for two triangles to be similar?

If two polygons are similar, what can be said about their areas?

What is the purpose of the Proportion Theorem?

If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

What is the condition for two polygons to be classified as similar?

What is the relationship between the corresponding sides of two similar triangles?

What is the property of proportion that states that wz = xy?

What is the condition for two triangles to be similar?

What does the Pythagorean theorem state?

What is the range of the regression coefficient (r)?

What is the formula for the area of a triangle?

What is the converse of the Pythagorean theorem?

What does a regression coefficient (r) close to -1 indicate?

What is the purpose of the linear regression line?

What is the equation of the linear regression line?

What is the condition for two polygons to be similar?

What can be inferred about the lengths of sides DE and BC if DE is drawn parallel to BC and DE is half the length of BC?

What is the purpose of using the distance formula and cosine rule to derive trigonometric identities?

What is the result of equating the two expressions KL^2 = (cos α - cos β)^2 + (sin α - sin β)^2 and KL^2 = 2 - 2 cos(α - β)?

What is the purpose of rewriting the angle as a difference in the derivation of cos(α + β)?

What is the result of substituting the even-odd identities into the derivation of cos(α + β)?

What is the purpose of using the unit circle in deriving trigonometric identities?

What is the trigonometric identity derived from equating KL^2 = (cos α - cos β)^2 + (sin α - sin β)^2 and KL^2 = 2 - 2 cos(α - β)?

What is the radius of a circle with equation (x - 3)^2 + (y + 2)^2 = 16?

If C(a, b) is the center of a circle, what is the equation of the tangent line to the circle at the point (a, b)?

What is the coordinate of the center of a circle with equation x^2 + y^2 + 6x - 4y + 4 = 0?

Which of the following equations represents a circle with center at (2, -3) and radius 5?

If a circle has equation x^2 + y^2 - 4x - 6y + 8 = 0, what is the radius?

What is the equation of the circle with center at (-1, 2) and radius 3?

If a circle has equation (x + 2)^2 + (y - 3)^2 = 16, what is the center?

Which formula represents the cosine of a sum correctly?

What is the equivalent expression for $ an(90^ ext{o} - eta)$?

What is the correct expression for the sine of double angle?

Which identity correctly represents the cosine of a difference?

Which of the following is NOT a form of the cosine of double angle?

What method is used to find the general solution to a trigonometric equation?

What does the CAST diagram help determine?

Which formula correctly defines the sine of a sum?

In the context of solving trigonometric equations, what must be verified after finding potential solutions?

What is the correct general solution for the equation if \( an heta = x\)?

Which of the following scenarios indicates the use of the Sine Rule?

In the context of calculating areas of triangles, when should the Area Rule be used?

What is the outcome of applying the Cosine Rule, given that two sides and the included angle are known?

Which formula correctly applies to find the height of a pole, given the relevant triangle and angles?

How can the height of a building be determined using the Sine Rule?

If the equation for the area of triangle ABC is \( ext{Area} = \frac{1}{2}ab\sin C\), which conditions must be met to use this equation?

In three-dimensional problems, what is the first step recommended before applying any rules?

A circle has the equation ( (x - 3)^2 + (y + 2)^2 = 25 ). A tangent line touches the circle at the point ( (7, 1) ). What is the equation of the tangent line?

Given two triangles, ( \triangle ABC ) and ( \triangle DEF ), with ( \angle A = \angle D ), ( \angle B = \angle E ), and ( \angle C = \angle F ). If ( AB = 12 ), ( BC = 8 ), ( AC = 10 ), and ( DE = 18 ), what is the length of ( EF )?

A line segment ( \overline{DE} ) is drawn parallel to the base ( \overline{BC} ) of triangle ( \triangle ABC ). If ( AD = 6 ), ( DB = 4 ), and ( AE = 9 ), what is the length of ( EC )?

In a triangle ( \triangle ABC ), ( \overline{DE} ) is a line segment parallel to ( \overline{BC} ), intersecting ( \overline{AB} ) at ( D ) and ( \overline{AC} ) at ( E ). If ( AD = 5 ), ( DB = 3 ), and ( AE = 8 ), what is the length of ( EC )?

A circle has the equation ( (x + 2)^2 + (y - 1)^2 = 16 ). A tangent line touches the circle at the point ( (2, 5) ). What is the gradient of the tangent line?

A circle with center ( (2, -3) ) passes through the point ( (5, 1) ). What is the equation of the tangent line to the circle at the point ( (5, 1) )?

Two similar triangles have corresponding sides in the ratio 3:5. If the area of the smaller triangle is 27 square units, what is the area of the larger triangle?

A line segment is drawn parallel to the base of a triangle, dividing the two sides proportionally. The length of one segment of a side is 6 units, and the length of the corresponding segment on the other side is 8 units. If the length of the whole side is 15 units, what is the length of the whole other side?

Two triangles are similar. The area of the smaller triangle is 16 square units, and the area of the larger triangle is 64 square units. If the perimeter of the smaller triangle is 12 units, what is the perimeter of the larger triangle?

A circle has a radius of 5 units and a tangent line touches the circle at the point ( (3, 4) ). What is the gradient of the radius drawn to this point of tangency?

Given a linear regression equation with a gradient of 0.6 and standard deviations of the (x)-values and (y)-values of 2.5 and 3.0 respectively, what is the value of the linear correlation coefficient (r)?

Two variables have a correlation coefficient of -0.75. What can be concluded about the relationship between the two variables?

Consider a circle centered at the origin with radius (r). What is the equation of the circle if it passes through the point ((3, 4))?

Given (cos (α - β) = 0.8) and (sin (α - β) = 0.6), what is the value of (tan (α - β))?

What is the simplified expression for (sin (α + β)cos (α - β) + cos (α + β)sin (α - β))?

In a right triangle with one angle measuring 60 degrees, the side opposite the 60-degree angle measures 10 units. What is the length of the hypotenuse?

A line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. This is a statement of which geometric theorem?

What is the value of (cos (α - β)) if (sin α = \frac{3}{5}), (cos α = \frac{4}{5}), (sin β = \frac{5}{13}) and (cos β = \frac{12}{13})?

What is the value of (r) if the linear correlation coefficient is 0.8 and the gradient of the least squares regression line is 0.5, given that the standard deviation of the (x)-values is 2?

If a circle has a diameter of 10 units, what is the equation of the circle if its center is at ((-2, 3))?

If the gradient of the radius is 2/3 and the gradient of the tangent is m, what is the value of m?

In a triangle, if a line is drawn parallel to one side, it divides the other two sides in a ratio of 3:5. If the length of one of the divided sides is 12, what is the length of the other divided side?

What is the equation of the tangent to a circle at the point of tangency (x₁, y₁) if the center of the circle is (a, b) and the radius is r?

If two triangles have their corresponding sides in a ratio of 2:3, what can be concluded about their angles?

What is the property of proportion that states wx = zy?

In a triangle, if a line is drawn parallel to one side, it divides the other two sides in a ratio of 2:5. If the length of one of the divided sides is 10, what is the length of the other divided side?

Given a right-angled triangle with sides of length 5, 12, and 13 units, what is the area of the triangle?

What is the next step after finding the center of the circle in determining the equation of the tangent?

If two polygons are similar, what can be said about their areas?

What is the purpose of the Basic Proportionality Theorem in geometry?

If two triangles are equiangular, what can be inferred about their sides?

In triangle ABC, DE is drawn parallel to BC, with D on AB and E on AC. If AD = 4, DB = 6, and AE = 5, what is the length of EC?

Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, and DE = 9, what is the length of EF?

Triangle ABC has a base BC of length 12 and a height of 8. A line segment DE is drawn parallel to BC, with D on AB and E on AC, such that DE is half the length of BC. What is the area of triangle ADE?

Given that triangle ABC is similar to triangle DEF, with AB = 6, BC = 8, and DE = 9, what is the ratio of the area of triangle ABC to the area of triangle DEF?

In triangle ABC, D is the midpoint of AB and E is the midpoint of AC. If BC = 10, what is the length of DE?

If two triangles have all their corresponding angles equal, but their corresponding sides are not proportional, what can you conclude?

A line segment DE is drawn parallel to BC in triangle ABC, with D on AB and E on AC. If AD = 3, DB = 6, and AE = 4, what is the length of EC?

Triangle ABC is similar to triangle DEF. If AB = 5, BC = 7, and DE = 10, what is the perimeter of triangle DEF?

A line segment DE is drawn parallel to BC in triangle ABC, with D on AB and E on AC. If AD = 4, DB = 6, and AE = 5, what is the ratio of the area of triangle ADE to the area of triangle ABC?

Triangle ABC has a base BC of length 10 and a height of 6. A line segment DE is drawn parallel to BC, with D on AB and E on AC, such that DE is one-third the length of BC. What is the area of triangle ADE?

Given a triangle with base length of 10 units and height of 6 units, and another triangle with the same height and base length of 5 units, what is the ratio of the area of the larger triangle to the area of the smaller triangle?

If a line segment connects the midpoints of two sides of a triangle, what conclusion can be made about the relationship between the line segment and the third side of the triangle?

In a triangle ( riangle ABC ), a line segment ( DE ) is drawn parallel to ( BC ) with ( D ) on ( AB ) and ( E ) on ( AC ). If ( AD = 4 ) units, ( DB = 6 ) units, and ( AE = 5 ) units, what is the length of ( EC )?

A kite has diagonals of lengths 12 units and 8 units. What is the area of the kite?

If two triangles have their corresponding sides in proportion, what can be concluded about their angles?

A trapezium has bases of lengths 10 units and 16 units, and its height is 8 units. What is the area of the trapezium?

Two triangles are said to be similar if they have the same shape but different sizes. What is the relationship between the areas of two similar triangles?

Given a triangle ( riangle ABC ), with ( D ) on ( AB ) and ( E ) on ( AC ), such that ( DE \parallel BC ). If ( AD = 3 ) units, ( DB = 5 ) units, and ( AE = 4 ) units, what is the length of ( EC )?

Two triangles are equal in area. If they share the same base, what conclusion can be made about their heights?

In a triangle, if a line segment is drawn connecting the midpoints of two sides, what is the relationship between the length of this line segment and the length of the third side?

Given the equation of a circle: ( x^2 + y^2 - 6x + 4y - 12 = 0 ), find the equation of the tangent line at the point ( (7, 1) ) on the circle.

A circle is tangent to the x-axis at the point ( (3, 0) ) and passes through the point ( (0, 4) ). Find the equation of the circle.

Two circles, ( C_1 ) and ( C_2 ), are tangent externally at the point ( (2, 3) ). The center of ( C_1 ) is at ( (0, 0) ), and the radius of ( C_1 ) is ( 5 ). Find the equation of circle ( C_2 ).

What is the interpretation of a correlation coefficient of -0.8?

Consider a circle with equation ( (x - 2)^2 + (y + 1)^2 = 9 ). The line ( y = mx + c ) is a tangent to this circle. Determine the possible values of ( m ) and ( c ).

What is the formula for the linear correlation coefficient r?

The equation of a circle is ( x^2 + y^2 + 8x - 10y + 16 = 0 ). Find the equation of the tangent line to the circle at the point ( (1, 5) ).

What is the equation of a circle with center at the origin and radius r?

What is the cosine of the difference of two angles α and β?

Given the circle with equation ( (x + 3)^2 + (y - 2)^2 = 25 ), find the equation of the tangent line that is parallel to the line ( 2x - y = 1 ).

A circle has center ( (1, -2) ) and passes through the point ( (4, 1) ). Find the equation of the tangent line to the circle at the point ( (4, 1) ).

What is the sine of the sum of two angles α and β?

The equation of a circle is ( x^2 + y^2 - 4x + 6y - 3 = 0 ). Find the coordinates of the point on the circle where the tangent line is parallel to the line ( 3x - 4y = 1 ).

What is the interpretation of a correlation coefficient of 0.4?

What is the symmetry of a circle with center at the origin?

What is the calculation of the regression line?

What is the purpose of using a calculator for linear regression?

What is the range of the correlation coefficient r?

What is the conclusion derived from equating the distance formula and the cosine rule regarding the cosine of the difference of two angles?

Which reasoning is applied to derive $ ext{cos}( heta + eta)$ using the negative angle identity?

What is simplified from the expression using even-odd identities when calculating $ ext{cos}( heta + eta)$?

What is the key relationship derived between two points on the unit circle using the distance formula?

Which identity represents the sine of the sum of two angles derived from the sine difference formula?

What common misunderstanding might arise regarding the cosine and sine functions when applying angle identities?

In a triangle ABC, given that a = 7, b = 5, and angle C = 60 degrees, what is the length of side c? (Use the Cosine Rule)

What is the expression for the sine of a sum of two angles?

Which identity represents the cosine of a difference of two angles?

What is the equivalent expression for $ an(2 heta)$ using double angle formulas?

In the identity for the cosine of a sum, $ heta + eta$, which component changes?

Which formula would you use to simplify $ an(2 heta)$?

What derives the sine of a double angle?

Which representation correctly uses the Pythagorean identity for sine and cosine?

What is the period of the sine and cosine functions?

Which angle identity is applicable when determining complementary angles?

Which expression equals $sin(90^ ext{o} - heta)$?