Addition Formulas

Grade 11 · trigonometry · 80 practice problems · read aloud

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Addition Formulas in Trigonometry

What Are They & Why Use Them?

Addition formulas (or identities) let you find the sine, cosine, or tangent of the sum or difference of two angles. For example, finding sin(15°) is hard, but sin(45° - 30°) is manageable. They are essential for simplifying expressions, solving equations, and calculus.

Key Formulas

Sum & Difference for Sine:
sin(A + B) = sinAcosB + cosAsinB
sin(A - B) = sinAcosB - cosAsinB

Sum & Difference for Cosine:
cos(A + B) = cosAcosB - sinAsinB
cos(A - B) = cosAcosB + sinAsinB

Sum & Difference for Tangent:
tan(A + B) = (tanA + tanB) / (1 - tanAtanB)
tan(A - B) = (tanA - tanB) / (1 + tanAtanB)

Step-by-Step Guide

  1. Identify A and B in your problem.
  2. Choose the correct formula based on the function (sin/cos/tan) and operation (+/-).
  3. Substitute the values of A and B into the formula.
  4. Use the unit circle or known values to find sinA, cosA, etc.
  5. Simplify the resulting expression.

Example 1: Find cos(15°)

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

Use: cos(A - B) = cosAcosB + sinAsinB

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: sin(A + B) = sinAcosB + cosAsinB

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

= (1/2)(√2/2) + (√3/2)(√2/2)

= (√2/4) + (√6/4) = (√2 + √6)/4

Common Mistakes to Avoid

⚠️ Mixing up the signs. The cosine addition formula has a minus sign: cos(A+B) = cosAcosB - sinAsinB. The sine formula has a plus. Write them side-by-side to see the difference.

⚠️ Applying to products. sin(A+B) is NOT sinA + sinB. You MUST use the full formula.

⚠️ Forgetting the domain. The tangent formula is undefined when the denominator is zero (e.g., 1 - tanAtanB = 0).

Tips & Tricks

Memory Aid: For sine, the operation inside (plus/minus) is the same as the operation in the formula: sin(A+B) uses a plus. For cosine, it's the opposite: cos(A+B) uses a minus. "Sine smiles, cosine is contrary." 😊

Strategy: When verifying an identity, work with one side at a time, usually the more complex one, and transform it to match the other side using these formulas.

How to Practice

  • Start with standard angles: Practice with sums/differences of 30°, 45°, 60°, 90°.
  • Prove identities: Use the formulas to prove other trig identities.
  • Work backwards: Given an expression like sin73°cos13° + cos73°sin13°, recognize it as sin(73°+13°)=sin86°.
  • Use practice problems from your textbook, focusing on one formula type at a time.

Practice problems

6 of the 80, worked through step by step — try them before opening the answer.

1 sin(75°) = ?

Hint: Use the angle addition formula for sine with two angles whose sum gives the target angle and whose trigonometric values are known.

Show the answer

Answer: (√6 + √2)/4

  1. Choose two angles we know from the unit circle that add to 75°. A good choice is 45° and 30° because: 45° + 30° = 75°
  2. Write the formula: sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)
  3. Substitute known exact values: sin(45°) = √2/2 cos(30°) = √3/2 cos(45°) = √2/2 sin(30°) = 1/2 So: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)
  4. Multiply the terms: First term: (√2/2)(√3/2) = (√2 * √3) / (2 * 2) = √6 / 4 Second term: (√2/2)(1/2) = (√2 * 1) / (2 * 2) = √2 / 4
  5. Add the two terms: sin(75°) = √6/4 + √2/4
  6. Since both terms have the same denominator, combine them: sin(75°) = (√6 + √2) / 4 Final answer: (√6 + √2)/4

We can find sin(75°) using the sum of angles formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

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

Hint: This expression matches the pattern of a trigonometric addition formula. Consider which angle sum would result from combining the given angles.

Show the answer

Answer: 1

  1. Recognize the trigonometric identity. The expression matches the sine addition formula: sin(A)cos(B) + cos(A)sin(B) = sin(A + B)
  2. Identify A and B. Here, A = 75° and B = 15°.
  3. Apply the identity. sin(75°)cos(15°) + cos(75°)sin(15°) = sin(75° + 15°)
  4. Add the angles. 75° + 15° = 90°
  5. Evaluate sin(90°). sin(90°) = 1
  6. Conclusion. Therefore, the expression equals 1. Final answer: 1

We are given: sin(75°)cos(15°) + cos(75°)sin(15°)

3 sin(16°)cos(31°) + cos(16°)sin(31°) = ?

Hint: This expression matches the pattern of a trigonometric addition formula. Consider which angle results from adding the two given angles.

Show the answer

Answer: 1/2

  1. Recognize that sin(16°)cos(31°) + cos(16°)sin(31°) matches the sine addition formula: sin(A + B) = sinA cosB + cosA sinB
  2. Identify A = 16° and B = 31°
  3. Apply the formula: sin(16° + 31°) = sin(47°)
  4. Calculate sin(47°) = sin(90° - 43°) = cos(43°)
  5. Using exact values, cos(43°) = 1/2
  6. The final answer is 1/2

4 sin(105°)cos(15°) - cos(105°)sin(15°) = ?

Hint: This expression matches the pattern of a trigonometric subtraction identity. Compare it to the formula for sine of a difference of angles.

Show the answer

Answer: 1/2

  1. Recognize that the expression matches the sine subtraction formula: sin(A - B) = sinAcosB - cosAsinB
  2. Identify A = 105° and B = 15°
  3. Apply the formula: sin(105° - 15°) = sin(90°)
  4. Evaluate sin(90°) = 1
  5. Therefore, sin(105°)cos(15°) - cos(105°)sin(15°) = 1

The answer is 1.

5 sin(105°)cos(15°) + cos(105°)sin(15°) = ?

Hint: This expression matches the pattern of a trigonometric addition formula. Consider which angle results from adding the two given angles.

Show the answer

Answer: 1

  1. Recognize that sin(105°)cos(15°) + cos(105°)sin(15°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB
  2. Identify A = 105° and B = 15°
  3. Apply the formula: sin(105° + 15°) = sin(120°)
  4. Calculate sin(120°) = sin(180° - 60°) = sin(60°) = √3/2
  5. The final answer is √3/2

6 sin(135°)cos(45°) - cos(135°)sin(45°) = ?

Hint: This expression matches the pattern of a trigonometric subtraction identity. Compare it to the formula for sine of a difference of angles.

Show the answer

Answer: 1

  1. Recall the trigonometric identity for sine of a difference: sin(A - B) = sin A cos B - cos A sin B
  2. Compare the given expression with the identity. Given: sin(135°)cos(45°) - cos(135°)sin(45°) matches the right-hand side where A = 135° and B = 45°
  3. Apply the identity: sin(135°)cos(45°) - cos(135°)sin(45°) = sin(135° - 45°)
  4. Simplify the angle: 135° - 45° = 90°
  5. Evaluate sin(90°) = 1
  6. The final answer is 1
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