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
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
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
- Find cos 15° using the addition formula (45° − 30°).
- Find tan 105° (60° + 45°).
- When sin α = 3/5 and cos β = 5/13 (α, β are acute angles), find sin(α+β).
Answers and Explanations
- Answer:(√6+√2)/4
Explanation: cos15° = cos45°cos30° + sin45°sin30° = (√6+√2)/4. - 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. - 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。