Trigonometry Advanced Trigonometric Concepts 4

Question 1

Which of the following is a solution to sin(2x) = sinx on [0, 2π)?

Accepted Answer

x = 0, π/3, π, 5π/3

Answer

sin(2x) = sinx → 2sinx cosx - sinx = 0 → sinx(2cosx - 1) = 0, giving x = 0, π/3, π, 5π/3.

Question 2

The inverse function arctan(x) has what range?

Accepted Answer

(-π/2, π/2)

Answer

arctan(x) is restricted to (-π/2, π/2) to ensure it is a one-to-one function.

Question 3

What is the sum-to-product form of sin A + sin B?

Accepted Answer

2 sin((A+B)/2) cos((A-B)/2)

Answer

The sum-to-product identity is sinA + sinB = 2 sin((A+B)/2) cos((A-B)/2).

Question 4

If cot θ = -√3 and θ is in the second quadrant, what is θ?

Accepted Answer

5π/6

Answer

cot θ = -√3 means tan θ = -1/√3; in the second quadrant this gives θ = 5π/6.

Question 5

What does the product-to-sum identity give for cos A · cos B?

Accepted Answer

(1/2)[cos(A-B) + cos(A+B)]

Answer

The product-to-sum formula is cosA cosB = (1/2)[cos(A - B) + cos(A + B)].

Trigonometry Practice Test