Trigonometry Advanced Trigonometric Concepts 5

Question 1

Which identity is used to verify that sin⁴θ - cos⁴θ = sin²θ - cos²θ?

Accepted Answer

Difference of squares: a² - b² = (a-b)(a+b)

Answer

Factor sin⁴θ - cos⁴θ = (sin²θ - cos²θ)(sin²θ + cos²θ) = (sin²θ - cos²θ)(1) using difference of squares and the Pythagorean identity.

Question 2

If sinα = 3/5 and cosβ = 5/13 (both in QI), what is sin(α + β)?

Accepted Answer

56/65

Answer

sin(α+β) = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65.

Question 3

What transformation maps y = sinx to y = sin(-x)?

Accepted Answer

Reflection over the y-axis

Answer

Replacing x with -x reflects the graph over the y-axis; since sin is odd, sin(-x) = -sinx confirms this is an x-axis reflection too, but the direct transformation is reflection over the y-axis.

Question 4

What is the general solution of sin x = √2/2?

Accepted Answer

x = π/4 + 2kπ or x = 3π/4 + 2kπ

Answer

sin x = √2/2 at x = π/4 and x = 3π/4 in [0,2π), so the general solution adds 2kπ to each.

Question 5

The Law of Sines states that in any triangle, a/sinA equals which of the following?

Accepted Answer

b/sinB

Answer

The Law of Sines: a/sinA = b/sinB = c/sinC — each side divided by the sine of its opposite angle is constant.

Trigonometry Practice Test

Trigonometry Practice Test

Trigonometry Advanced Trigonometric Concepts 5