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Sum, Difference & Double-Angle Formulas

You’ll be able to

Trigonometric functions are not linear

sin(A + B) is not sin A + sin B. Test it: sin(π/2 + π/2) = sin π = 0, while sin(π/2) + sin(π/2) = 2. The correct expansions mix sines with cosines, and that mixing is exactly what makes the identities useful — it lets you build unfamiliar angles out of familiar ones.

Sum and difference formulas
sin(A ± B) = sin A cos B ± cos A sin B · cos(A ± B) = cos A cos B ∓ sin A sin B
Note the sign reversal in the cosine formula: cos(A + B) uses a **minus**. This is the detail worth over-rehearsing.
Worked example

Find the exact value of cos(15°).

  1. 1.Write 15° as a difference of familiar angles: 45° − 30°.
  2. 2.cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°.
  3. 3.Substitute: (√2/2)(√3/2) + (√2/2)(1/2).
  4. 4.= √6/4 + √2/4 = (√6 + √2)/4.
  5. 5.Numerically: (2.449 + 1.414)/4 ≈ 0.966, and cos 15° ≈ 0.966. ✓
Answer: cos(15°) = (√6 + √2)/4 ≈ 0.966. The plus sign appears because cos(A − B) uses addition — the reverse of cos(A + B). Checking against a decimal value is worth the ten seconds it takes.

Double angles are just B = A

Set B = A in the sum formulas and everything follows. sin(2A) = sin A cos A + cos A sin A = 2 sin A cos A. And cos(2A) = cos²A − sin²A, which the Pythagorean identity converts into two more forms: 1 − 2sin²A and 2cos²A − 1. Three expressions for the same thing, and choosing the right one is what makes a simplification collapse in one step instead of five.

Double-angle formulas
sin 2A = 2 sin A cos A · cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
Pick the cos 2A form that matches what you already have. If the expression contains sin²A, use 1 − 2sin²A; if it contains cos²A, use 2cos²A − 1.
Checkpoint

Simplify 2 sin(3x) cos(3x).

Worked example

Given sin A = 3/5 with A in quadrant II, find sin(2A) and cos(2A).

  1. 1.Find cos A. From sin²A + cos²A = 1, cos²A = 1 − 9/25 = 16/25, so cos A = ±4/5.
  2. 2.A is in quadrant II, where cosine is negative, so cos A = −4/5.
  3. 3.sin 2A = 2 sin A cos A = 2(3/5)(−4/5) = −24/25.
  4. 4.cos 2A = 1 − 2sin²A = 1 − 2(9/25) = 1 − 18/25 = 7/25.
  5. 5.Consistency check: sin²2A + cos²2A = 576/625 + 49/625 = 625/625 = 1. ✓
Answer: sin(2A) = −24/25 and cos(2A) = 7/25. Since sin 2A < 0 and cos 2A > 0, the doubled angle lies in quadrant IV — doubling an angle moves it to a different quadrant, so quadrant information about A does not transfer to 2A.
Watch out

The quadrant of A determines the sign of cos A, and getting it wrong flips the sign of sin 2A. Always resolve the quadrant before using the Pythagorean identity — √(16/25) has two square roots and only one is right.

Where these identities earn their keep

Three uses recur. Exact values for angles like 15° or 75° that are sums or differences of the special angles. Simplification, collapsing a two-term expression into one — which is what makes an equation solvable. And solving equations containing both sin x and sin 2x, where the double-angle formula rewrites everything in terms of a single angle so the substitution methods from the previous lesson apply.

Checkpoint

To solve sin(2x) = sin(x), what is the useful first step?

Answer the 2 checkpoints as you read.

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