🎯 Learning Objectives
- Why sin θ = cos(90° − θ): the cofunction identity.
- How to use this rule to convert any sine into a cosine and vice versa.
- How the SAT disguises this rule inside word problems.
- Quick checks that catch sign errors.
- 10 SAT-style practice questions.
1. Why It Works: The Geometry
In a right triangle, the two acute angles always add up to 90°. (The third angle is the right angle, taking the remaining 90°.) So if one angle is θ, the other is 90° − θ. They are complementary.
Now consider the SAME triangle, but pick a different acute angle. The side that was 'opposite' for one angle is 'adjacent' for the other. That swap is exactly what turns sine into cosine.
The cofunction identity
sin θ = cos (90° − θ)
cos θ = sin (90° − θ)
tan θ = cot (90° − θ)
💡 In Words
The sine of an angle equals the cosine of the angle's complement. The 'co' in cosine literally stands for 'complement'.
2. The Quick Translation Strategy
STEP 1: Identify the two angles in the equation
STEP 2: Set up: x + y = 90°
STEP 3: Solve the resulting linear equation
📝 Worked Example
If sin (3x − 10)° = cos (x + 20)°, find x.
Because the angles are complementary: (3x − 10) + (x + 20) = 90.
4x + 10 = 90 → 4x = 80 → x = 20.
3. Common SAT Disguises
The SAT rarely asks about complementary angles directly. Instead, you'll see one of these three disguises.
Disguise 1: Same Value, Different Function
'If sin A = 4/5, what is cos B if A and B are complementary?'
Answer: 4/5. The sine of one angle equals the cosine of its complement.
Disguise 2: Triangle Side Ratios
'In right triangle ABC with right angle at C, sin A = 3/5. What is cos B?'
Since A and B are complementary, cos B = sin A = 3/5.
Disguise 3: Algebraic Equation
'If sin (2x)° = cos (50)°, find x.'
Set 2x + 50 = 90 → x = 20.
4. A Useful Related Identity
The Pythagorean identity
sin² θ + cos² θ = 1
This is just the Pythagorean theorem in disguise. In a right triangle with hypotenuse 1, the legs are sin θ and cos θ, so their squares add to 1.
💡 Pro Tip: Use It to Find Missing Values
If the SAT tells you sin θ = 3/5 and asks for cos θ, use sin² + cos² = 1:
cos² θ = 1 − 9/25 = 16/25, so cos θ = 4/5.
(Assuming θ is acute; otherwise cos θ could also be −4/5.)
5. Common Mistakes
⚠️ Common Mistakes
- Adding the angles to 180° instead of 90°. Complementary means 90°.
- Forgetting that the identity applies only to complementary angles, not just any two angles.
- Mixing up sin² θ with sin(θ²). These are very different.
- Confusing 'supplementary' (sum to 180°) with 'complementary' (sum to 90°).
6. Strategies
⚡ Spot sin = cos? Complementary angles.
Set the angles' sum to 90° and solve in one step.
⚡ sin of one acute angle = cos of the other
In any right triangle, the two non-right angles are complements. Use this to skip Pythagorean work.
⚡ Use sin² + cos² = 1 for missing values
If you know one, you can get the other.
⚡ For 'sin 60° = cos ___', the answer is 30°
Because 60° + 30° = 90°.
⚡ Check units: always use degrees or radians consistently
Mixing them up is a silent killer on test day.
7. Summary
📌 Key Takeaways
- Core identity: sin θ = cos (90° − θ).
- When sin A = cos B: A + B = 90°.
- In a right triangle: the two acute angles are always complementary.
- Pythagorean identity: sin² θ + cos² θ = 1.
- Complementary = 90°, supplementary = 180° (don't confuse them).
8. Practice: 10 Questions
Q1. If sin(3x)° = cos(60)°, find x.
A) 10 B) 20 C) 30 D) 60
Hint: 3x + 60 = 90 → x = 10.
Q2. If sin A = 0.6 and A + B = 90°, what is cos B?
A) 0.4 B) 0.6 C) 0.8 D) 0
Hint: cos B = sin A = 0.6.
Q3. If sin(x + 20)° = cos(x + 10)°, find x.
A) 20 B) 30 C) 40 D) 50
Hint: (x+20) + (x+10) = 90 → 2x+30 = 90 → x = 30.
Q4. In a right triangle, the acute angles are A and B. If sin A = 5/13, find cos B.
A) 5/13 B) 12/13 C) 13/5 D) 13/12
Hint: cos B = sin A = 5/13.
Q5. If sin θ = 3/5, what is cos θ (assume θ is acute)?
A) 3/5 B) 4/5 C) 5/3 D) 5/4
Hint: sin² + cos² = 1 → cos² = 16/25 → cos = 4/5.
Q6. sin 45° = cos ___?
A) 30° B) 45° C) 60° D) 90°
Hint: 45 + 45 = 90.
Q7. If sin(2x)° = cos(x + 30)°, find x.
A) 10 B) 20 C) 30 D) 40
Hint: 2x + (x+30) = 90 → 3x = 60 → x = 20.
Q8. If cos A = 0.8 and A + B = 90°, what is sin B?
A) 0.2 B) 0.6 C) 0.8 D) 1
Hint: sin B = cos A = 0.8.
Q9. In right triangle ABC with C = 90°, sin A = 7/25. cos B?
A) 7/25 B) 24/25 C) 25/7 D) 1/7
Hint: A and B are complementary → cos B = sin A = 7/25.
Q10. sin² 30° + cos² 30° = ?
A) 0 B) 1/2 C) 1 D) 2
Hint: Pythagorean identity: always 1.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) 10 | 3x + 60 = 90. |
Q2. | B) 0.6 | cos B = sin A. |
Q3. | B) 30 | 2x + 30 = 90. |
Q4. | A) 5/13 | cos B = sin A. |
Q5. | B) 4/5 | sin² + cos² = 1. |
Q6. | B) 45° | 45 + 45 = 90. |
Q7. | B) 20 | 3x = 60. |
Q8. | C) 0.8 | sin B = cos A. |
Q9. | A) 7/25 | Complementary → values match. |
Q10. | C) 1 | Pythagorean identity. |