A student computes sin(45° + 30°) by writing sin(45°) + sin(30°) = 0.707 + 0.500 = 1.207. What is wrong, and what is the correct value?
ANothing is wrong — distributing sine over addition is valid when the angles sum to 75°
BThe student used the wrong identity; the correct answer is (√6 + √2)/4 ≈ 0.966, not 1.207
CThe student should have multiplied the sines instead: sin(45°) × sin(30°)
DThe student's method works only when both angles are from the standard unit circle
Sine is not a linear function — it does not distribute over addition. The sum and difference identity gives sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4 ≈ 0.966. A quick sanity check: sin(75°) must be less than 1 since 75° ≠ 90°, so 1.207 is already impossible. The misconception that sin(A+B) = sinA + sinB is the single most common error in trigonometry.
Question 2 Multiple Choice
Which of the following correctly states the cosine difference formula?
Acos(A − B) = cos A cos B − sin A sin B
Bcos(A − B) = cos A cos B + sin A sin B
Ccos(A − B) = sin A cos B − cos A sin B
Dcos(A − B) = cos A sin B + sin A cos B
The cosine formulas follow the pattern 'cosine changes sign': cos(A+B) = cosAcosB − sinAsinB (minus) and cos(A−B) = cosAcosB + sinAsinB (plus). The minus in the sum becomes a plus in the difference. This is opposite to the sine formulas, where the sign in the identity matches the sign of the argument. A useful check: cos(0) = cos(A−A) = cos²A + sin²A = 1 ✓.
Question 3 True / False
The formula cos(A + B) = cos A cos B + sin A sin B is correct.
TTrue
FFalse
Answer: False
This is false — the correct formula has a minus sign: cos(A+B) = cosAcosB − sinAsinB. The sign pattern for cosine is that the sum formula has a minus and the difference formula has a plus. This is opposite to sine (where sum → plus, difference → minus). A useful check: cos(60°) = cos(30°+30°) = cos²30° − sin²30° = 3/4 − 1/4 = 1/2 ✓. Using plus would give 3/4 + 1/4 = 1, which would mean cos(60°) = 1 — clearly wrong.
Question 4 True / False
The formulas for sin(A + B) and sin(A − B) differ only in the sign between their two terms.
TTrue
FFalse
Answer: True
True. sin(A+B) = sinAcosB + cosAsinB, and sin(A−B) = sinAcosB − cosAsinB. The two terms are identical; only the connecting sign changes. This pattern (sign matches the ± in the argument) applies to sine but not cosine, where the signs are inverted.
Question 5 Short Answer
Why can't you compute sin(A + B) simply by adding sin A and sin B? Explain using a specific counterexample.
Think about your answer, then reveal below.
Model answer: Because sine is a nonlinear function — it doesn't distribute over addition. Counterexample: sin(30° + 60°) = sin(90°) = 1, but sin(30°) + sin(60°) = 0.5 + 0.866 = 1.366 ≠ 1. The correct formula requires cross terms: sin(A+B) = sinAcosB + cosAsinB. These cross terms reflect how the angles interact geometrically on the unit circle.
The linearity property f(A+B) = f(A)+f(B) holds for functions like f(x) = kx, but not for trigonometric functions. The correct formula has four terms (two products), capturing the interaction between the two angles. The counterexample with 30°+60°=90° is particularly clean because sin(90°)=1 is a known value, making the error in the naive approach obvious.