Sum Difference Formulas

Grade 11 · trigonometry · 93 practice problems · read aloud

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Sum and Difference Formulas

What Are They & Why Learn Them?

Sum and Difference Formulas let you find the sine, cosine, or tangent of an angle that is the sum or difference of two known angles. For example, you can find sin(15°) by thinking of it as sin(45° - 30°). This is incredibly useful for simplifying expressions and solving trigonometric equations that don't involve standard unit circle angles.

The Core Formulas

Sine Formulas:

  • sin(A + B) = sin A cos B + cos A sin B
  • sin(A - B) = sin A cos B - cos A sin B

Cosine Formulas:

  • cos(A + B) = cos A cos B - sin A sin B
  • cos(A - B) = cos A cos B + sin A sin B

Tangent Formulas:

  • tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
  • tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Step-by-Step Guide

  1. Identify A and B: Determine the two angles being added or subtracted.
  2. Choose the correct formula: Match the operation (sum or difference) and the trig function.
  3. Substitute values: Plug the known values for sin, cos, or tan of A and B into the formula.
  4. Simplify: Perform the arithmetic 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) = cos A cos B + sin A sin B

Substitute: 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° = 30° + 45°. So, A=30°, B=45°.

Use the sum formula for sine: sin(A + B) = sin A cos B + cos A sin B

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

= (1/2)(√2/2) + (√3/2)(√2/2)

= (√2/4) + (√6/4) = (√2 + √6)/4

Common Mistakes ⚠️

  • Mixing up the signs: The cosine sum formula has a minus sign, while the cosine difference has a plus. It's the opposite of what many intuitively guess!
  • Using the wrong function order: For sine, it's always "sin cos ± cos sin". Don't mix this up.
  • Forgetting the denominator for tangent: The formula for tan(A+B) is a fraction. A common error is to forget the "1 - tan A tan B" part.

Tips & Tricks

  • Memory Aid: For sine, the operation (plus/minus) in the formula is the same as the operation between the angles: sin(A + B) = sinAcosB + cosAsinB. For cosine, it's the opposite: cos(A + B) = cosAcosB - sinAsinB.
  • Focus on Sine and Cosine: You can always find tangent using tanθ = sinθ/cosθ if you forget the specific tangent formulas.
  • Check Your Quadrants: Ensure the sign of your final answer makes sense based on the quadrant of the original angle.

How to Practice

1. Drill the Fundamentals: Start by finding exact values for sums/differences of common angles (like 15°, 75°, 105°).

2. Verify Known Identities: Use the formulas to prove that sin(90°)=1 or cos(180°)=-1.

3. Apply to Equations: Solve equations like sin(x+π/3) = 1/2.

4. Mix and Match: Create practice problems that require you to use multiple formulas in one problem.

Practice problems

6 of the 93, worked through step by step — try them before opening the answer.

1 sin(165°) = ?

Hint: Express 165° as a sum or difference of two angles whose sine and cosine values are known exactly.

Show the answer

Answer: (√6 - √2)/4

  1. Express 165° as a sum or difference of known angles. A good choice is 165° = 120° + 45°.
  2. Apply the sine sum formula: sin(A + B) = sin A cos B + cos A sin B.
  3. Substitute A = 120° and B = 45°: sin(165°) = sin(120°)cos(45°) + cos(120°)sin(45°).
  4. Find the exact values: sin(120°) = √3/2, cos(45°) = √2/2, cos(120°) = -1/2, sin(45°) = √2/2.
  5. Substitute the values: sin(165°) = (√3/2)(√2/2) + (-1/2)(√2/2) = (√6/4) - (√2/4).
  6. Combine the terms: (√6/4) - (√2/4) = (√6 - √2)/4. The exact value of sin(165°) is (√6 - √2)/4.

2 cos(165°) = ?

Hint: Use the cosine sum formula with angles that have known exact values. Consider how 165° can be expressed as a sum of two special angles.

Show the answer

Answer: -√6/4 - √2/4

  1. Express 165° as a sum of angles with known cosine and sine values: 165° = 120° + 45°
  2. Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB
  3. Substitute A = 120° and B = 45°: cos(165°) = cos(120°)cos(45°) - sin(120°)sin(45°)
  4. Use exact values: cos(120°) = -1/2, cos(45°) = √2/2, sin(120°) = √3/2, sin(45°) = √2/2
  5. Substitute the values: cos(165°) = (-1/2)(√2/2) - (√3/2)(√2/2)
  6. Simplify: cos(165°) = -√2/4 - √6/4
  7. Combine terms: cos(165°) = -√6/4 - √2/4

The answer is -√6/4 - √2/4.

3 cos(195°) = ?

Hint: Consider expressing the angle as a sum or difference of two special angles whose trigonometric values you know exactly.

Show the answer

Answer: -(√6+√2)/4

  1. Express 195° as a sum of two special angles: 195° = 180° + 15°.
  2. Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB. Let A = 180° and B = 15°.
  3. Substitute known values: cos(180°) = -1, sin(180°) = 0, cos(15°) = (√6+√2)/4, sin(15°) = (√6-√2)/4.
  4. Calculate: cos(195°) = (-1) * ((√6+√2)/4) - (0) * ((√6-√2)/4) = -(√6+√2)/4. The exact value of cos(195°) is -(√6+√2)/4.

4 sin(255°) = ?

Hint: Express 255° as the sum of two special angles (180° and 75°) or as 180° + 75°, then apply the sine sum formula sin(A+B) = sinA cosB + cosA sinB. Use known exact values for sin(180°), cos(180°), sin(75°), and cos(75°).

Show the answer

Answer: -(√6+√2)/4

  1. Express 255° as a sum of two special angles: 255° = 180° + 75°.
  2. Apply the sine sum formula: sin(A+B) = sinA cosB + cosA sinB, with A = 180°, B = 75°.
  3. sin(180°) = 0, cos(180°) = -1.
  4. Find exact values for sin(75°) and cos(75°). Use 75° = 45° + 30°. sin(75°) = sin(45°+30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6+√2)/4. cos(75°) = cos(45°+30°) = cos45°cos30° - sin45°sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = √6/4 - √2/4 = (√6-√2)/4.
  5. Substitute into the sum formula: sin(255°) = sin(180°)cos(75°) + cos(180°)sin(75°) = (0)*((√6-√2)/4) + (-1)*((√6+√2)/4) = -(√6+√2)/4. The exact value of sin(255°) is -(√6+√2)/4.

5 tan(285°) = ?

Hint: Express 285° as the sum of two angles whose tangent values you know exactly, such as 240° and 45°. Then apply the tangent sum formula: tan(A+B) = (tan A + tan B) / (1 - tan A tan B). Remember the exact values for tan(240°) and tan(45°).

Show the answer

Answer: -(2+√3)

  1. Express 285° as a sum: 285° = 240° + 45°.
  2. Apply the tangent sum formula: tan(A+B) = (tan A + tan B) / (1 - tan A tan B), with A = 240°, B = 45°.
  3. Find exact values: tan(240°) = tan(180°+60°) = tan(60°) = √3 (since tangent has period 180°). tan(45°) = 1.
  4. Substitute into the formula: tan(285°) = (√3 + 1) / (1 - √3 * 1) = (√3 + 1) / (1 - √3).
  5. Rationalize the denominator: Multiply numerator and denominator by the conjugate (1 + √3): = (√3 + 1)(1 + √3) / ((1 - √3)(1 + √3)) = (√3 + 1)(√3 + 1) / (1 - 3) = ( (√3)^2 + 2√3 + 1 ) / (-2) = (3 + 2√3 + 1) / (-2) = (4 + 2√3) / (-2) = -2 - √3. The exact value of tan(285°) is -(2+√3).

6 cos(285°) = ?

Hint: Express 285° as the sum of two special angles (e.g., 180° + 105° or 225° + 60°), then apply the cosine sum formula. You will need exact values for the component angles, which may themselves require sum/difference formulas.

Show the answer

Answer: (√6 - √2)/4

  1. Express 285° as a sum of two special angles: 285° = 225° + 60°.
  2. Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB, with A = 225°, B = 60°.
  3. Find exact values: cos(225°) = -√2/2, sin(225°) = -√2/2, cos(60°) = 1/2, sin(60°) = √3/2.
  4. Substitute: cos(285°) = (-√2/2)(1/2) - (-√2/2)(√3/2) = -√2/4 + (√2·√3)/4 = -√2/4 + √6/4 = (√6 - √2)/4. The exact value of cos(285°) is (√6 - √2)/4.
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