Sine and Cosine of Complementary Angles — SAT Math Guide

Q: What You Need to Know

• Complementary angles add up to 90° (π/2 radians) • For complementary angles A and B: sin A = cos B and cos A = sin B • The cofunction identity: sin θ = cos(90° - θ) and cos θ = sin(90° - θ) • Common complementary angle pairs: 30°-60°, 45°-45° • This relationship works for any angle measurement (de

Sine and cosine of complementary angles SAT questions test your understanding of the special relationship between these trigonometric functions. Complementary angles are two angles that add up to 90°, and they have a unique property that makes SAT problems much easier once you know the pattern. This topic appears in the Geometry and Trigonometry domain, typically showing up in 1-2 questions per Digital SAT. You'll find these concepts straightforward once you master the core identity.

What You Need to Know

Complementary angles add up to 90° (π/2 radians)

For complementary angles A and B: sin A = cos B and cos A = sin B

The cofunction identity: sin θ = cos(90° - θ) and cos θ = sin(90° - θ)

Common complementary angle pairs: 30°-60°, 45°-45°

This relationship works for any angle measurement (degrees or radians)

📐 KEY FORMULA: sin θ = cos(90° - θ) and cos θ = sin(90° - θ)

💡 PRO TIP: When you see sin 30° = cos 60°, remember they're complementary angles (30° + 60° = 90°)!

How to Solve Sine and Cosine of Complementary Angles SAT Problems

Example Question 1 — Medium Difficulty

If sin 35° = k, what is the value of cos 55°?

A) -k

B) k

C) 1 - k

D) 1/k

Solution:

Step 1: Recognize that 35° and 55° are complementary angles (35° + 55° = 90°)

Step 2: Apply the cofunction identity: sin 35° = cos(90° - 35°) = cos 55°

Step 3: Since sin 35° = k, then cos 55° = k

✅Answer: B — Because 35° and 55° are complementary angles, sin 35° equals cos 55°.

Example Question 2 — Hard Difficulty

In triangle ABC, angle C is a right angle. If sin A = 3/5 and angle A is acute, what is the value of cos B?

A) 3/5

B) 4/5

C) 5/3

D) 5/4

Solution:

Step 1: Since triangle ABC has a right angle at C, angles A and B are complementary (A + B = 90°)

Step 2: Use the complementary angle relationship: sin A = cos B

Step 3: Since sin A = 3/5, then cos B = 3/5

Step 4: Verify this makes sense (both sine and cosine values are between 0 and 1 for acute angles)

✅Answer: A — In a right triangle, the sine of one acute angle equals the cosine of the other acute angle.

Common SAT Math Mistakes to Avoid

❌Mistake: Confusing complementary (90°) with supplementary (180°) angles

✅Fix: Remember "C" for Complementary = 90° Corner of a right angle

❌Mistake: Thinking sin 30° = cos 30° because they look similar

✅Fix: Use the complementary relationship: sin 30° = cos 60°, not cos 30°

❌Mistake: Applying the identity incorrectly with obtuse angles

✅Fix: The basic identity works for any angle, but watch signs in different quadrants

❌Mistake: Forgetting to check if angles actually add to 90°

✅Fix: Always verify that the angles are truly complementary before applying the cofunction identity

Practice Question — Try It Yourself

If cos 40° = m, which expression is equivalent to sin 50°?

A) -m

B) m

C) 1 - m²

D) √(1 - m²)

Show Answer

Answer: B — Since 40° + 50° = 90°, these are complementary angles, so sin 50° = cos 40° = m.

Key Takeaways for the SAT

Complementary angles always add up to 90°, and their sine and cosine values are equal

The cofunction identity is your best friend: sin θ = cos(90° - θ)

SAT math problems often give you one trig value and ask for its complement

In right triangles, the two acute angles are automatically complementary

This relationship works in both degrees and radians on the Digital SAT