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Sum and Difference Identities

A free Trigonometry lesson from the “Trig Identities” unit, with a worked example and practice problems including step-by-step solutions.

The sum and difference identities let you find exact trig values for angles that aren't on the unit circle by writing them as combinations of 30, 45, and 60 degrees. The key formulas are sin(a+/-b) = sin(a)cos(b) +/- cos(a)sin(b), cos(a+/-b) = cos(a)cos(b) -/+ sin(a)sin(b), and tan(a+/-b) = (tan(a) +/- tan(b))/(1 -/+ tan(a)tan(b)). Watch the cosine sign flip: the sign inside cos(a+b) becomes minus on the right side.

What you'll learn

Why it matters: Sum and difference identities are the foundation of how signals combine in AC circuits, audio mixing, and physics: adding two waves of different phase produces a predictable new wave that these formulas describe exactly.

Worked example

Problem. Find the exact value of sin(75) using a sum identity.

  1. Write 75 = 45 + 30, so sin(75) = sin(45)cos(30) + cos(45)sin(30)
  2. Substitute: (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = sqrt(6)/4 + sqrt(2)/4
  3. Combine over 4: (sqrt(6)+sqrt(2))/4

Answer: (sqrt(6)+sqrt(2))/4

Practice problems

1. Which expansion is correct for sin(a + b)?

Choices: sin(a)cos(b) + cos(a)sin(b) · sin(a)cos(b) - cos(a)sin(b) · cos(a)cos(b) + sin(a)sin(b) · sin(a)sin(b) + cos(a)cos(b)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which expansion is correct for sin(a + b)?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The sine sum identity keeps the same sign as the angle
  4. sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sin(a)cos(b) + cos(a)sin(b)

2. To find an exact value for cos(15) using a difference identity, which split should you use?

Choices: 45 - 30 · 60 - 30 · 90 - 75 · 20 - 5

Show solution
  1. Warm-up: First identify exactly what the question is asking: To find an exact value for cos(15) using a difference identity, which split should you use?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. You need two angles from {30, 45, 60} whose difference is 15
  4. 45 - 30 = 15, and both are known unit-circle angles
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 45 - 30

3. Expand cos(a - b) as a formula in terms of sin and cos of a and b.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Expand cos(a - b) as a formula in terms of sin and cos of a and b.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The cosine identity flips the sign relative to the angle
  4. cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
  5. Check the result by substituting or estimating: the response should match cos(a)cos(b) + sin(a)sin(b) and make sense in the original problem.

Answer: cos(a)cos(b) + sin(a)sin(b)

4. Find the exact value of cos(15). Give the answer as a single fraction.

Show solution
  1. cos(15) = cos(45-30) = cos(45)cos(30) + sin(45)sin(30)
  2. (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = sqrt(6)/4 + sqrt(2)/4
  3. Combine: (sqrt(6)+sqrt(2))/4

Answer: (sqrt(6)+sqrt(2))/4

5. Find the exact value of sin(15). Give the answer as a single fraction.

Show solution
  1. sin(15) = sin(45-30) = sin(45)cos(30) - cos(45)sin(30)
  2. (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2) = sqrt(6)/4 - sqrt(2)/4
  3. Combine: (sqrt(6)-sqrt(2))/4

Answer: (sqrt(6)-sqrt(2))/4

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