Trigonometry Flashcards

Which expression is equivalent to cos(2x)?

Answer: 1 - 2sin²(x)

cos(2x) = 1 - 2sin²(x) is one of the standard double-angle identities.

Solve for x in [0, 2π]: 2cos(x) - 1 = 0.

Answer: x = π/3 or x = 5π/3

cos(x) = 1/2 gives x = π/3 and x = 5π/3 in [0, 2π].

What is tan(π/4 + π/4) using the addition formula?

Answer: undefined

tan(π/2) is undefined because it equals tan(π/4 + π/4) and cos(π/2) = 0.

Express sin(75°) using the sum formula.

Answer: (√6 + √2)/4

sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4.

For which values of x is the equation sin²(x) = sin(x) satisfied in [0, 2π]?

Answer: x = 0, π/2, π, 2π

sin²x = sin x ⟹ sin x(sin x - 1) = 0 ⟹ sin x = 0 or sin x = 1, giving x = 0, π, 2π, π/2.

The function f(x) = sin(x) + cos(x) can be written in the form R·sin(x + φ). What is R?

Answer: √2

R = √(1² + 1²) = √2 for f(x) = sin(x) + cos(x).

What is the exact value of sin(π/12)?

Answer: (√6 - √2)/4

sin(π/12) = sin(15°) = sin(45° - 30°) = (√6 - √2)/4.