Sum & Difference Identities

Section 7.2 — finding exact values for non-special angles by splitting them into special angles

MAT 172 — Week 3
sin(75°) isn't on your special angles chart — but 75° = 45° + 30° and both of those are. Sum and difference identities let you compute exact values for any angle that can be built from 30°, 45°, 60°, 90°, and 180°.
Six identities to know. The sum and difference formulas for sin, cos, and tan. Notice the pattern in the signs — sin and tan have matching signs (+ for sum, − for difference), but cos has the OPPOSITE sign.
sin(A + B)
sin A cos B + cos A sin B
sin(A − B)
sin A cos B − cos A sin B
cos(A + B)
cos A cos B − sin A sin B
cos(A − B)
cos A cos B + sin A sin B
tan(A + B)
(tan A + tan B) / (1 − tan A tan B)
tan(A − B)
(tan A − tan B) / (1 + tan A tan B)
⚠️
The sign trick — cos is backwards
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The hardest part is keeping track of signs. For sin and tan, the sign in the formula matches the sign of the original: sin(A + B) has a plus, sin(A − B) has a minus. But for cos, it's flipped: cos(A + B) has a MINUS, cos(A − B) has a PLUS.
Memory trick: sin and tan match their sign. cos flips it.
Or: cos(A + B) = coscos − sinsin (both terms negative for sum)
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sin(A + B) ≠ sin A + sin B
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This is the most common mistake. Trig functions do NOT distribute over addition. sin(45° + 30°) ≠ sin(45°) + sin(30°). Check: sin(75°) ≈ 0.966, but sin(45°) + sin(30°) = 0.707 + 0.5 = 1.207. That's not even possible (sine is always ≤ 1). You MUST use the sum formula.
✗ sin(A + B) ≠ sin A + sin B
✓ sin(A + B) = sin A cos B + cos A sin B
Which angles can you build from special angles? Common buildable angles: 15° = 45° − 30° 75° = 45° + 30° 105° = 60° + 45° 165° = 135° + 30° 195° = 180° + 15° 255° = 270° − 15° −π/12 = π/6 − π/4 7π/12 = π/3 + π/4 Special angles: 0°, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2), 120°, 135°, 150°, 180°, 270°, 360° Any combination of these using + or − can be computed with sum/difference formulas
Pattern: Split the angle into two special angles. Identify A and B. Plug into the formula. Simplify using special angle values.
sin(75°) — exact value
1
Split: 75° = 45° + 30°. Use sin(A + B) = sin A cos B + cos A sin B
2
= sin(45°)cos(30°) + cos(45°)sin(30°)
3
= (√2/2)(√3/2) + (√2/2)(1/2)
4
= √6/4 + √2/4
sin(75°) = (√6 + √2) / 4 ≈ 0.966
cos(π/12) — exact value [π/12 = π/4 − π/6 = 45° − 30°]
1
π/12 = π/4 − π/6. Use cos(A − B) = cos A cos B + sin A sin B
2
= cos(π/4)cos(π/6) + sin(π/4)sin(π/6)
3
= (√2/2)(√3/2) + (√2/2)(1/2)
4
= √6/4 + √2/4
cos(π/12) = (√6 + √2) / 4
tan(105°) — exact value [105° = 60° + 45°]
1
Use tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
2
= (tan 60° + tan 45°)/(1 − tan 60° · tan 45°)
3
= (√3 + 1)/(1 − √3 · 1) = (√3 + 1)/(1 − √3)
4
Rationalize: multiply by (1 + √3)/(1 + √3) → (√3 + 1 + 3 + √3)/(1 − 3) = (4 + 2√3)/(−2)
tan(105°) = −2 − √3
Given values — no angles given
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sin A = 3/5 (QI), cos B = −5/13 (QII). Find sin(A + B).
1
Find missing values. cos A = 4/5 (QI, positive). sin B = 12/13 (QII, positive).
2
sin(A + B) = sin A cos B + cos A sin B
3
= (3/5)(−5/13) + (4/5)(12/13)
4
= −15/65 + 48/65
sin(A + B) = 33/65
Co-function identities from difference formulas. The co-function identities (sin θ = cos(90°−θ), etc.) are actually special cases of the difference formulas. Here's why they're true.
Deriving co-function identities from difference formulas sin θ = cos(90° − θ) Start with cos(A − B) = cos A cos B + sin A sin B Let A = 90°, B = θ: cos(90° − θ) = cos 90°·cos θ + sin 90°·sin θ = (0)·cos θ + (1)·sin θ = sin θ ✓ The "co" in cosine literally means complement — cosine of an angle equals the sine of its complement (90° − θ). This derivation proves it using the difference formula for cosine. All co-function identities sin θ = cos(π/2 − θ) cos θ = sin(π/2 − θ) tan θ = cot(π/2 − θ) cot θ = tan(π/2 − θ) sec θ = csc(π/2 − θ) csc θ = sec(π/2 − θ) All provable from difference formulas
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Negative angle identities (even/odd)
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Also provable from difference formulas — let A = 0:

sin(−θ) = −sin(θ) — sine is an odd function (symmetric about origin)
cos(−θ) = cos(θ) — cosine is an even function (symmetric about y-axis)
tan(−θ) = −tan(θ) — tangent is odd

These come from cos(0 − θ) = cos 0·cos θ + sin 0·sin θ = cos θ (since cos 0=1, sin 0=0). And sin(0 − θ) = sin 0·cos θ − cos 0·sin θ = −sin θ.
Pick two angles A and B. See the sum and difference identities verified in real time — the formula result vs the direct calculation.
45°
30°
A + B
75°
A − B
15°
Check: sin(A+B)
0.966

Blue = sin(A+B) computed directly. Orange = formula result. They always match ✓

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All six formulas

sin(A±B)
sin A cos B ± cos A sin B
cos(A+B)
cos A cos B − sin A sin B
cos(A−B)
cos A cos B + sin A sin B
tan(A±B)
(tan A ± tan B)/(1 ∓ tan A tan B)

Key vocabulary

Sum formula
Identity for trig of (A + B). Note: cos(A+B) has a MINUS in the middle, not plus.
Difference formula
Identity for trig of (A − B). cos(A−B) has a PLUS — opposite of what you'd expect.
Exact value
Answer expressed with radicals and fractions, not decimals. sin(75°) = (√6+√2)/4 is exact.
Co-function identity
sin θ = cos(90°−θ) etc. Provable from the difference formula for cosine.

Exam reminders

cos(A+B) has MINUS. cos(A−B) has PLUS. They're backwards from what you expect.
sin(A+B) ≠ sin A + sin B. Always use the formula.
For "given values" problems: find missing sides with Pythagorean theorem, then check the quadrant for signs.
15° = 45°−30°. 75° = 45°+30°. 105° = 60°+45°. Build non-special angles from the ones you know.
tan formula has 1 ∓ tan A tan B in denominator — the sign flips from the numerator.