TMUA Trigonometry 3

Question 1

Which expression is equivalent to cos(2x)?

Accepted Answer

1 - 2sin²(x)

Answer

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

Question 2

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

Accepted Answer

x = π/3 or x = 5π/3

Answer

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

Question 3

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

Accepted Answer

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Answer

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

Question 4

Express sin(75°) using the sum formula.

Accepted Answer

(√6 + √2)/4

Answer

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

Question 5

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

Accepted Answer

x = 0, π/2, π, 2π

Answer

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

(TMUA) Test of Mathematics for University Admission Practice Test