Sum Difference Formulas

Grade 12 · trigonometry · 100 practice problems · read aloud

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Sum and Difference Formulas

What Are They & Why Useful?

Sum and Difference Formulas let you find exact trigonometric values for angles that are sums or differences of common angles (like 30°, 45°, 60°). They are essential for simplifying expressions, solving equations, and proving identities.

Core Formulas

For Sine:

  • sin(A+B) = sinA cosB + cosA sinB
  • sin(A−B) = sinA cosB − cosA sinB

For Cosine:

  • cos(A+B) = cosA cosB − sinA sinB
  • cos(A−B) = cosA cosB + sinA sinB

For Tangent:

  • tan(A+B) = (tanA + tanB) / (1 − tanA tanB)
  • tan(A−B) = (tanA − tanB) / (1 + tanA tanB)

Step-by-Step Guide

  1. Identify A and B from the given sum or difference.
  2. Choose the correct formula (sine, cosine, or tangent).
  3. Substitute the values of A and B into the formula.
  4. Find the exact values of sin, cos, tan for A and B.
  5. Simplify the expression to get your final answer.

Example 1: Find cos(15°)

We know 15° = 45° − 30°. So, A=45°, B=30°.

Use the difference formula for cosine: cos(A−B) = cosA cosB + sinA sinB

cos(45°−30°) = cos45°cos30° + sin45°sin30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6/4) + (√2/4) = (√6 + √2)/4

Example 2: Find sin(75°)

75° = 45° + 30°. So, A=45°, B=30°.

Use the sum formula for sine: sin(A+B) = sinA cosB + cosA sinB

sin(45°+30°) = sin45°cos30° + cos45°sin30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6/4) + (√2/4) = (√6 + √2)/4

⚠️ Common Mistakes

  • Sign Errors: The cosine sum formula has a minus sign: cos(A+B) = cosAcosB − sinAsinB. Mixing this up is the #1 error.
  • Function Mix-up: Using sine formula for a cosine problem. Double-check the function you're solving for.
  • Misidentifying A and B: Ensure you correctly break down the angle into known values.

💡 Tips & Tricks

  • Memory Aid: For sine, the operation (plus/minus) inside the formula is the same as the operation outside: sin(A+B) = sinAcosB + cosAsinB.
  • For cosine, it's the opposite: cos(A+B) = cosAcosB − sinAsinB.
  • Always look for angles that sum to or differ from 0°, 30°, 45°, 60°, 90°.

Practice Suggestions

  • Start by finding exact values for sin(15°), cos(105°), tan(75°).
  • Prove identities using the formulas to change the left side into the right side.
  • Solve equations like sin(x+π/4) = 1/2.
  • Mix in problems where you have to find sin(A+B) given sinA and cosB in different quadrants.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 cos(75°) = ?

Hint: Consider expressing the angle as a sum or difference of two common angles whose cosine and sine values are known.

Show the answer

Answer: (√6 - √2)/4

  1. Recognize that 75° can be written as 45° + 30°.
  2. Apply the cosine sum formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
  3. Substitute A = 45° and B = 30°: cos(75°) = cos(45°)cos(30°) - sin(45°)sin(30°).
  4. Use known values: cos(45°) = √2/2, cos(30°) = √3/2, sin(45°) = √2/2, sin(30°) = 1/2.
  5. Substitute the values: cos(75°) = (√2/2)(√3/2) - (√2/2)(1/2).
  6. Simplify the expression: cos(75°) = (√6/4) - (√2/4).
  7. Combine the terms: cos(75°) = (√6 - √2)/4.

The answer is (√6 - √2)/4.

2 sin(105°) = ?

Hint: This angle can be expressed as the sum of two common angles whose trigonometric values are well-known.

Show the answer

Answer: (√6 + √2)/4

  1. Recognize that 105° can be written as 60° + 45°.
  2. Apply the sine sum formula: sin(A + B) = sinAcosB + cosAsinB.
  3. Substitute A = 60° and B = 45°: sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°).
  4. Use known values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2.
  5. Substitute the values: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2).
  6. Simplify the expression: sin(105°) = (√6/4) + (√2/4).
  7. Combine the terms: sin(105°) = (√6 + √2)/4.

The answer is (√6 + √2)/4.

3 cos(105°) = ?

Hint: This angle can be expressed as a sum of two common angles. Consider which trigonometric identity applies to the cosine of a sum.

Show the answer

Answer: (√2 - √6)/4

  1. Recognize that 105° can be written as 60° + 45°.
  2. Apply the cosine sum formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
  3. Substitute A = 60° and B = 45°: cos(105°) = cos(60°)cos(45°) - sin(60°)sin(45°).
  4. Use known values: cos(60°) = 1/2, cos(45°) = √2/2, sin(60°) = √3/2, sin(45°) = √2/2.
  5. Substitute the values: cos(105°) = (1/2)(√2/2) - (√3/2)(√2/2).
  6. Simplify the expression: cos(105°) = (√2/4) - (√6/4).
  7. Combine the terms: cos(105°) = (√2 - √6)/4.

The answer is (√2 - √6)/4.

4 sin(15°)cos(75°) + cos(15°)sin(75°) = ?

Hint: This expression matches the pattern of a sum formula from trigonometry. Consider which trigonometric identity applies when you see sinAcosB + cosAsinB.

Show the answer

Answer: 1

  1. Recognize that sin(15°)cos(75°) + cos(15°)sin(75°) matches the pattern of the sine sum formula: sin(A+B) = sinAcosB + cosAsinB
  2. Apply the identity: sin(15°)cos(75°) + cos(15°)sin(75°) = sin(15° + 75°)
  3. Add the angles: 15° + 75° = 90°
  4. Evaluate: sin(90°) = 1

The answer is 1.

5 sin(50°)cos(20°) + cos(50°)sin(20°) = ?

Hint: This expression follows the pattern of a trigonometric sum identity. Identify which identity matches the structure sinAcosB + cosAsinB.

Show the answer

Answer: √3/2

  1. Recognize that sin(50°)cos(20°) + cos(50°)sin(20°) matches the sine sum formula: sin(A+B) = sinAcosB + cosAsinB
  2. Apply the identity: sin(50°)cos(20°) + cos(50°)sin(20°) = sin(50° + 20°)
  3. Add the angles: 50° + 20° = 70°
  4. Evaluate sin(70°). Since 70° = 90° - 20°, sin(70°) = cos(20°)
  5. cos(20°) = √3/2 (exact value)

The answer is √3/2.

6 sin(π/3)cos(π/4) - cos(π/3)sin(π/4) = ?

Hint: This expression follows the pattern of a trigonometric difference formula. Identify which identity matches the structure sinAcosB - cosAsinB.

Show the answer

Answer: (√6 - √2)/4

  1. Recognize that sin(π/3)cos(π/4) - cos(π/3)sin(π/4) matches the sine difference formula: sin(A - B) = sinAcosB - cosAsinB
  2. Apply the identity: sin(π/3)cos(π/4) - cos(π/3)sin(π/4) = sin(π/3 - π/4)
  3. Subtract the angles: π/3 - π/4 = (4π/12 - 3π/12) = π/12
  4. Evaluate sin(π/12). Note that π/12 = 15° and can be written as 45° - 30°
  5. Apply the sine difference formula again: sin(45° - 30°) = sin45°cos30° - cos45°sin30°
  6. Substitute known values: sin45° = √2/2, cos30° = √3/2, cos45° = √2/2, sin30° = 1/2
  7. Calculate: (√2/2)(√3/2) - (√2/2)(1/2) = (√6/4) - (√2/4)
  8. Combine terms: (√6 - √2)/4

The answer is (√6 - √2)/4.

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