7.2Using Sum and Difference Formulas
Apply cos(α ± β) = cosα cosβ ∓ sinα sinβ, sin(α ± β) = sinα cosβ ± cosα sinβ, and tan(α ± β) formulas to find exact values and verify identities.
Sum and difference identities let you find exact trig values at non-standard angles like 15° or 75°. They are also the foundation for the double-angle and half-angle identities used throughout calculus.
How can you find the exact value of a trig function at an angle that is not on the standard unit circle, by expressing it as a sum or difference of familiar angles?
The sum and difference formulas allow you to evaluate trig functions at angles formed by adding or subtracting two known angles. The key formulas are:
Memory trick for cosine
Cosine uses the opposite sign: cos(α + β) uses −, and cos(α − β) uses +. Sine uses the same sign throughout.
Strategy: Express the target angle as a sum or difference of angles whose trig values you know exactly: 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, and their radian equivalents.
Find the exact value of sin(75°).
Write 75° = 45° + 30°.
Apply sin(α + β) = sinα cosβ + cosα sinβ:
sin(75°) = sin45° cos30° + cos45° sin30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
Find the exact value of cos(15°).
Write 15° = 45° − 30°.
Apply cos(α − β) = cosα cosβ + sinα sinβ:
cos(15°) = cos45° cos30° + sin45° sin30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
Find the exact value of tan(π/12).
Note π/12 = π/4 − π/6 (since 3π/12 − 2π/12 = π/12).
Apply tan(α − β) = (tanα − tanβ)/(1 + tanα tanβ):
tan(π/12) = (tan(π/4) − tan(π/6))/(1 + tan(π/4)·tan(π/6))
= (1 − 1/√3)/(1 + 1·1/√3)
= (1 − 1/√3)/(1 + 1/√3)
Multiply numerator and denominator by √3: (√3 − 1)/(√3 + 1)
Rationalize: (√3 − 1)²/((√3)² − 1²) = (3 − 2√3 + 1)/2 = (4 − 2√3)/2
Prove: cos(π/2 − θ) = sinθ
Apply cos(α − β) with α = π/2, β = θ:
cos(π/2 − θ) = cos(π/2)cosθ + sin(π/2)sinθ
= 0 · cosθ + 1 · sinθ
= sinθ ✓
Given sinα = 3/5 (α in QI) and cosβ = −5/13 (β in QII), find sin(α + β).
Find cosα: cos²α = 1 − (3/5)² = 1 − 9/25 = 16/25 → cosα = 4/5 (QI, positive).
Find sinβ: sin²β = 1 − (−5/13)² = 1 − 25/169 = 144/169 → sinβ = 12/13 (QII, positive).
Apply sin(α + β) = sinα cosβ + cosα sinβ:
= (3/5)(−5/13) + (4/5)(12/13)
= −15/65 + 48/65
Find the exact value of sin(105°).
Hint: Write 105° = 60° + 45° and apply the sine sum formula.
Find the exact value of cos(π/12).
Hint: Write π/12 = π/4 − π/6 and apply the cosine difference formula.
Prove: sin(π − θ) = sinθ
Hint: Apply sin(α − β) with α = π, β = θ. Use sin(π) = 0 and cos(π) = −1.
Given cosα = −4/5 (α in QIII) and sinβ = 5/13 (β in QI), find cos(α − β).
Hint: Find sinα (negative in QIII) and cosβ (positive in QI), then apply the cosine difference formula.
Find the exact value of tan(165°).
Hint: Write 165° = 120° + 45° and apply the tangent sum formula. tan(120°) = −√3.
Interactive Practice — 5 Questions
Which formula gives cos(α + β)?
sin(75°) expressed as a sum is:
cos(π/2 − θ) equals:
To find sin(α + β), you need:
The exact value of cos(15°) is:
Common Mistakes
Writing cos(α + β) = cosα + cosβ (distributing cosine over addition).
cos(α + β) = cosα cosβ − sinα sinβ. Trig functions do NOT distribute over addition.
Using the wrong sign in the cosine formula: writing cos(α + β) = cosα cosβ + sinα sinβ.
Cosine uses the opposite sign: cos(α + β) uses −, and cos(α − β) uses +.
Forgetting to find all four values (sinα, cosα, sinβ, cosβ) before applying the formula.
Always determine all four values first, paying attention to the quadrant to get the correct sign.
Choosing the wrong quadrant sign for cosα or sinβ when given limited information.
Use the quadrant information given. In QII, cosine is negative; in QIII, both sine and cosine are negative.
Math Tips
Memorize the six formulas as three pairs. For sine: same sign throughout. For cosine: opposite sign. For tangent: same sign on top, opposite on bottom.
The cofunction identities (sin(π/2 − θ) = cosθ, etc.) are special cases of the difference formulas with α = π/2.
When finding exact values, always check whether the target angle can be written as a sum or difference of 30°, 45°, or 60° (or their multiples).
After computing, rationalize any denominator with a radical to put the answer in standard form.
For the tangent sum/difference formula, the denominator is 1 ∓ tanα tanβ — the sign is opposite to the angle operation.