ACT Math: Use Sum and Difference Angle Formulas Without Memorizing Them

sin(A+B)=sin(A)cos(B)+cos(A)sin(B). sin(A-B)=sin(A)cos(B)-cos(A)sin(B). cos(A+B)=cos(A)cos(B)-sin(A)sin(B). cos(A-B)=cos(A)cos(B)+sin(A)sin(B). These formulas appear when you need to evaluate a trig function at an unusual angle. Example: sin(75°)=sin(45°+30°). Apply the formula: sin(45°)cos(30°)+cos(45°)sin(30°)=(√2/2)(√3/2)+(√2/2)(1/2)=(√6+√2)/4. Without the formula, you would need a calculator. The pattern is: sin sums are "sine-cosine plus cosine-sine." Cosine sums are "cosine-cosine minus sine-sine."

Memory trick for signs: Sine sums keep the same sign (+ stays +, - stays -). Cosine sums flip the sign (+ becomes -, - becomes +). This rhyme helps you recall without memorizing four formulas separately.

Mistake 1: Misremembering the signs. For cosine, the sum formula has a minus (not a plus). Cosine is "weird" because the sign flips. Write it out or use the memory trick. Mistake 2: Forgetting which trig values to use. sin(75°)=sin(45°+30°) requires sin(45°), cos(30°), cos(45°), and sin(30°). Write these down before substituting; missing one ruins the entire calculation. Always write out the formula with the correct angle breakdown (A and B), then substitute the known trig values step-by-step.

Before you calculate, verify that you have identified A and B correctly and that you know the exact values of sin and cos for those angles (like sin(30°)=1/2). If the angles are non-standard (like sin(37°)), the ACT likely expects you to leave the answer in formula form or use a different approach.

Expression 1: sin(75°)=sin(45°+30°)=sin(45°)cos(30°)+cos(45°)sin(30°)=(√2/2)(√3/2)+(√2/2)(1/2)=√6/4+√2/4=(√6+√2)/4. Expression 2: cos(15°)=cos(45°-30°)=cos(45°)cos(30°)+sin(45°)sin(30°)=(√2/2)(√3/2)+(√2/2)(1/2)=(√6+√2)/4. Wait, this is the same as sin(75°) (they are complementary angles). Expression 3: sin(105°)=sin(60°+45°)=sin(60°)cos(45°)+cos(60°)sin(45°)=(√3/2)(√2/2)+(1/2)(√2/2)=(√6+√2)/4. For each expression, break the angle into two standard angles (30°, 45°, 60°), apply the formula, and substitute exact values.

After calculating, verify by checking whether the answer makes sense. sin(75°) should be close to sin(90°)=1 but a bit less. Your answer (√6+√2)/4 ≈ 0.966 matches this expectation.

Sum and difference angle formula questions appear 0-1 times per test, usually in the hardest ACT Math questions. These questions are not conceptually complex; they test whether you know the formulas and can substitute carefully. Once you master the formulas and the sign patterns, you solve these questions in under two minutes, earning points that many students skip because the formulas feel intimidating.

Spend 15 minutes this week learning the four formulas and the memory tricks. Practice evaluating 5 angles using sums and differences of standard angles. By test day, you will recognize when a sum-difference formula is needed and apply it without hesitation, unlocking points on some of the hardest questions on the test.