Trigonometry Trigonometric Identities 2

Question 1

Which identity correctly expresses cos(2θ) in terms of sin(θ) only?

Accepted Answer

cos(2θ) = 1 − 2sin²(θ)

Answer

The double-angle identity cos(2θ) = 1 − 2sin²(θ) is derived by substituting cos²θ = 1 − sin²θ into cos(2θ) = cos²θ − sin²θ.

Question 2

What is the simplified form of (1 − cos²θ) / sin θ?

Accepted Answer

sin θ

Answer

Since 1 − cos²θ = sin²θ by the Pythagorean identity, dividing by sin θ gives sin θ.

Question 3

Which of the following equals sin(A + B)?

Accepted Answer

sin A cos B + cos A sin B

Answer

The sum identity for sine is sin(A + B) = sin A cos B + cos A sin B.

Question 4

If tan θ = 3/4, what is the value of sin θ cos θ using identities?

Accepted Answer

12/25

Answer

With tan θ = 3/4, we get sin θ = 3/5 and cos θ = 4/5 in a 3-4-5 triangle, so sin θ cos θ = (3/5)(4/5) = 12/25.

Question 5

Which identity is used to prove that cot²θ + 1 = csc²θ?

Accepted Answer

Pythagorean identity sin²θ + cos²θ = 1 divided by sin²θ

Answer

Dividing sin²θ + cos²θ = 1 by sin²θ yields 1 + cot²θ = csc²θ, which rearranges to cot²θ + 1 = csc²θ.

Trigonometry Practice Test