Relationships between trig functions; symmetry practice questions.

For all values of θ, sin(π − θ) is equal to which of the following?

• −sin θ
• cos θ
• sin θ
• −cos θ

Answer: sin θ

The sine graph is symmetric about x = π/2, so supplementary angles have equal sines: sin(π − θ) = sin θ.

Which of the following is an odd function of θ?

• cos θ
• tan θ
• sec θ
• cos 2θ

Answer: tan θ

tan(−θ) = −tan θ, so tan is odd; cos, sec and cos 2θ are all even since cos(−θ) = cos θ.

Which expression is identically equal to cos θ for all values of θ?

• sin(π/2 − θ)
• sin(θ − π/2)
• −sin(π/2 − θ)
• sin(π − θ)

Answer: sin(π/2 − θ)

Sine and cosine are cofunctions of complementary angles: sin(π/2 − θ) = cos θ.

Given that sin θ = 3/5 and π/2 < θ < π, determine the value of cos θ.

• 4/5
• 3/4
• −3/4
• −4/5

Answer: −4/5

From sin²θ + cos²θ = 1, cos θ = ±4/5; in the second quadrant cosine is negative, so cos θ = −4/5.

For all θ where the expression is defined, tan(π + θ) is equal to

• −tan θ
• cot θ
• tan θ
• −cot θ

Answer: tan θ

The tangent function has period π, so tan(π + θ) = tan θ.

Which of the following expressions is identically equal to −cos θ?

• cos(−θ)
• cos(π − θ)
• sin(π − θ)
• cos(2π − θ)

Answer: cos(π − θ)

On the unit circle π − θ is the reflection of θ in the y-axis, which reverses the sign of the x-coordinate: cos(π − θ) = −cos θ. By contrast cos(−θ) and cos(2π − θ) both equal cos θ, and sin(π − θ) = sin θ.