Gr12 Mathematics: Ch 4.1 Compound Angle Identities

What is the formula for sin(α - β)?

cos α cos β + sin α sin β
cos α cos β - sin α sin β
sin α cos β - cos α sin β (correct)
sin α cos β + cos α sin β

What is the formula for cos(2α)?

cos^2 α + sin^2 α
2 sin α cos α
2 cos^2 α - 1 (correct)
cos^2 α - sin^2 α

What is the formula for sin(α + β)?

cos α cos β + sin α sin β
cos α cos β - sin α sin β
sin α cos β + cos α sin β (correct)
sin α cos β - cos α sin β

What is the formula for cos(α - β)?

cos α cos β - sin α sin β (A) Signup and view all the answers

What is the formula for sin(2α)?

2 sin α cos α (C) Signup and view all the answers

What is the formula for cos(α + β)?

cos α cos β - sin α sin β (B) Signup and view all the answers

What is the formula for the cosine of a difference?

$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ (A) Signup and view all the answers

Which method is used to derive the formula for $\cos(\alpha + \beta)$?

Using the negative angle identity (C) Signup and view all the answers

What is the formula for the sine of a sum?

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ (D) Signup and view all the answers

What is the formula for the cosine of a sum?

$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ (A) Signup and view all the answers

What is the purpose of deriving compound angle formulas?

To simplify trigonometric expressions involving sums and differences of angles (C) Signup and view all the answers

What is the formula for the sine of a difference?

$\sin(\alpha - \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ (A) Signup and view all the answers

Which of the following is not a method used to derive compound angle formulas?

Using the Pythagorean theorem (C) Signup and view all the answers

What is the advantage of using compound angle formulas?

They can be used to simplify complex trigonometric expressions (D) Signup and view all the answers

What is the correct application of the co-function identity in the derivation of the sine of a difference formula?

cos(90° - α) = sin α (B), sin(90° - α) = cos α (D) Signup and view all the answers

Which of the following is a correct application of the compound angle formula for sine?

sin(α + β) = sin α cos β + cos α sin β (B), sin(α - β) = sin α cos β - cos α sin β (D) Signup and view all the answers

What is the result of applying the double angle formula for sine to sin(4α)?

2 sin(2α) cos(2α) (B) Signup and view all the answers

Which of the following identities is used in the derivation of the cosine of a difference formula?

cos(α - β) = cos α cos β + sin α sin β (B) Signup and view all the answers

What is the difference between the formulas for cos(α - β) and cos(α + β)?

A negative sign is present in the former and a positive sign in the latter. (B) Signup and view all the answers

In deriving the formula for cos(α - β), which of the following is used?

The distance formula and cosine rule. (A) Signup and view all the answers

What is the purpose of using co-functions in the derivation of compound angle formulas?

To simplify the expressions (D) Signup and view all the answers

Which of the following formulas can be derived using the compound angle formula for cosine?

cos(2α) = cos^2 α - sin^2 α (A), cos(2α) = 1 - 2 sin^2 α (B), cos(2α) = 2 cos^2 α - 1 (C) Signup and view all the answers

What is the purpose of rewriting α + β as α - (-β) in deriving the formula for cos(α + β)?

To apply the cosine difference formula. (A) Signup and view all the answers

Which of the following is a consequence of the even-odd identities?

cos(-β) = cos(β) and sin(-β) = -sin(β). (D) Signup and view all the answers

What is the relation between the formulas for sin(α - β) and sin(α + β)?

They differ only in the sign of the cosine terms. (B) Signup and view all the answers

What is the underlying principle in deriving the formulas for compound angles?

The unit circle definition of trigonometric functions. (B) Signup and view all the answers

Why are compound angle formulas useful?

They simplify trigonometric expressions involving sums and differences of angles. (D) Signup and view all the answers

What is the significance of the method used to derive cos(α + β)?

It illustrates the importance of the even-odd identities. (C) Signup and view all the answers

If sin(α) = 3/5 and cos(β) = 4/5, what is the value of sin(α - β)?

7/25 (C) Signup and view all the answers

If cos(α) = 2/3 and sin(β) = 1/3, what is the value of cos(α + β)?

4/9 (C) Signup and view all the answers

If sin(α) = 1/2 and cos(α) = √3/2, what is the value of sin(2α)?

√3/2 (D) Signup and view all the answers

If cos(α) = 3/5 and sin(α) = 4/5, what is the value of cos(2α)?

-7/25 (C) Signup and view all the answers

What is the value of sin(α) cos(β) + cos(α) sin(β) in terms of compound angle formulas?

sin(α + β) (D) Signup and view all the answers

What is the value of cos(α) cos(β) - sin(α) sin(β) in terms of compound angle formulas?

cos(α + β) (D) Signup and view all the answers

What is the key step in deriving the formula for cos(α + β) using the negative angle identity?

Rewriting the angle as a difference α - (-β) (A) Signup and view all the answers

What is the purpose of using the distance formula and cosine rule in deriving the formula for cos(α - β)?

To equate two expressions for KL^2 (C) Signup and view all the answers

What is the common feature among the formulas for sin(α - β), sin(α + β), cos(α - β), and cos(α + β)?

They all involve the product of two trigonometric functions (C) Signup and view all the answers

What is the reason for the difference in signs between the formulas for cos(α - β) and cos(α + β)?

Due to the difference in the direction of the angles (C) Signup and view all the answers

What is the underlying principle behind the derivation of the formulas for compound angles?

The principle of equivalence of expressions (A) Signup and view all the answers

What is the role of the negative angle identity in deriving the formula for cos(α + β)?

To apply the cosine difference formula (D) Signup and view all the answers

What is the consequence of applying the even-odd identities in the derivation of the formulas for compound angles?

Simplification of the expressions (D) Signup and view all the answers

What is the key idea behind the derivation of the formulas for cos(α - β) and cos(α + β)?

Equating two expressions for KL^2 (D) Signup and view all the answers