Fix the mistake: Trigonometric Functions ② (Addition Theorems, Double and Half Angles)

Addition Formulas

The addition formulas are the most important group of formulas in trigonometric functions. Memorize them accurately and learn to use them effectively.

Addition Formulas (6 Formulas)

  • sin(α+β) = sin α cos β + cos α sin β
  • sin(α−β) = sin α cos β − cos α sin β
  • cos(α+β) = cos α cos β − sin α sin β
  • cos(α−β) = cos α cos β + sin α sin β
  • tan(α+β) = (tan α + tan β)/(1 − tan α tan β)
  • tan(α−β) = (tan α − tan β)/(1 + tan α tan β)
Example ①
Find sin 75° using the addition formula.
Answer:75° = 45° + 30°
sin 75° = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4
Example ②
When sin α = 1/2 and cos β = 1/3 (α, β are acute angles), find cos(α+β).
Answer:cos α = √3/2、sin β = 2√2/3。
cos(α+β) = (√3/2)(1/3) − (1/2)(2√2/3) = √3/6 − √2/3 = (√3 − 2√2)/6
Key Point
  • sin: use the cross pattern “sin cos ± cos sin”
  • cos: multiply like functions: “cos cos ∓ sin sin” (use a minus sign for addition)
  • tan: the numerator is a sum or difference, and the denominator is “1 ∓ product”

Practice Problems

  1. Find cos 15° using the addition formula (45° − 30°).
  2. Find tan 105° (60° + 45°).
  3. When sin α = 3/5 and cos β = 5/13 (α, β are acute angles), find sin(α+β).

Answers and Explanations

  1. Answer:(√6+√2)/4
    Explanation: cos15° = cos45°cos30° + sin45°sin30° = (√6+√2)/4.
  2. Answer:−2−√3
    Explanation: tan(60°+45°) = (√3+1)/(1−√3). Multiply the numerator and denominator by (1+√3): (4+2√3)/(−2) = −2−√3.
  3. Answer:56/65
    Explanation: cos α = 4/5、sin β = 12/13。sin(α+β) = (3/5)(5/13)+(4/5)(12/13) = 15/65+48/65 = 63/65。

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