IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~7 min read HL only

Compound Angle Formulae

Compound angle formulae let you expand sin, cos, and tan of a sum or difference of two angles. They unlock exact values for angles like 15°, 75°, 105° (by writing them as 45°±30° or 60°±45°), and they’re the foundation for everything that follows — double angle formulae, identities, and proofs.

📘 What you need to know

sin(A ± B) = sin A cos B ± cos A sin B (sign same on both sides).
cos(A ± B) = cos A cos B ∓ sin A sin B (sign flips — this is the trap).
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B) (top sign matches; bottom sign flips).
All three are in the formula booklet (tan version is in the HL section).
For exact values: split angles into multiples of 30°, 45°, 60° (or π/6, π/4, π/3).
For “given sin A, find sin(A+B)”: use the Pythagorean identity to recover the missing values first.
Used to derive the double angle formulas (next note) and to prove identities.

The three formulae

sin of a sum or difference sin(A ± B) = sin A cos B ± cos A sin B

cos of a sum or difference cos(A ± B) = cos A cos B ∓ sin A sin B

tan of a sum or difference tan(A ± B) = tan A ± tan B1 ∓ tan A tan B

The cos formula has a sign flip: cos(A + B) uses a minus, cos(A − B) uses a plus. This is the most common error in compound angle questions — slow down on the cos formula.

Exact values from compound angles

Any angle that’s a sum or difference of 30°, 45°, 60° can be evaluated exactly.

Angle	Split as
15°	45° − 30°
75°	45° + 30°
105°	60° + 45°
165°	120° + 45° or 180° − 15°
π/12 (15°)	π/4 − π/6
5π/12 (75°)	π/4 + π/6

Tip: in the formula booklet you’ll find the exact values for sin/cos/tan of 30°, 45°, 60°. Combine them with the compound angle formula and you get the exact value of any 15°-multiple.

🧭 Recipe — find an exact value with a compound angle

Split the angle as a sum or difference of 30°, 45°, 60° (or π/6, π/4, π/3).
Apply the relevant formula — sin, cos, or tan of A ± B.
Substitute the known exact values from the formula booklet.
Simplify by combining fractions; rationalise the denominator if needed.
Check sign by comparing with the calculator decimal value.

Worked examples

WE 1

Find sin 75° exactly

Without using a calculator, find the exact value of sin 75°. Give your answer in the form (a + √b)/c with integers a, b, c.

Step 1: Split 75° = 45° + 30° sin 75° = sin(45° + 30°) Step 2: Apply sin(A + B) = sin A cos B + cos A sin B = sin 45° cos 30° + cos 45° sin 30° Step 3: Substitute exact values = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 sin 75° = √6 + √24 decimal check: (√6+√2)/4 ≈ 0.9659 ≈ sin 75° ✓

WE 2

Find cos 105° exactly

Without using a calculator, find the exact value of cos 105°.

Step 1: Split 105° = 60° + 45° cos 105° = cos(60° + 45°) Step 2: Apply cos(A + B) = cos A cos B − sin A sin B = cos 60° cos 45° − sin 60° sin 45° Step 3: Substitute = (1/2)(√2/2) − (√3/2)(√2/2) = √2/4 − √6/4 cos 105° = √2 − √64 negative as expected: 105° is in Q2, where cos is negative ✓

WE 3

Find tan 15° exactly

Without using a calculator, find the exact value of tan 15°. Give your answer in simplified surd form.

Step 1: Split 15° = 45° − 30° tan 15° = tan(45° − 30°) Step 2: Apply tan(A − B) formula = (tan 45° − tan 30°) / (1 + tan 45° tan 30°) = (1 − 1/√3) / (1 + 1/√3) Step 3: Multiply top and bottom by √3 = (√3 − 1) / (√3 + 1) Step 4: Rationalise — multiply by (√3 − 1)/(√3 − 1) = (√3 − 1)² / ((√3)² − 1²) = (3 − 2√3 + 1) / (3 − 1) = (4 − 2√3) / 2 = 2 − √3 tan 15° = 2 − √3

WE 4

Find sin(A + B) and cos(A + B) from given values

Given that sin A = 3/5 with A acute, and cos B = −5/13 with B obtuse, find the exact values of sin(A + B) and cos(A + B).

Step 1: Find the missing values via Pythagoras A acute → cos A = +√(1 − 9/25) = 4/5 B obtuse → sin B = +√(1 − 25/169) = 12/13 Step 2: Apply sin(A + B) = sin A cos B + cos A sin B = (3/5)(−5/13) + (4/5)(12/13) = −15/65 + 48/65 = 33/65 Step 3: Apply cos(A + B) = cos A cos B − sin A sin B = (4/5)(−5/13) − (3/5)(12/13) = −20/65 − 36/65 = −56/65 sin(A + B) = 3365, cos(A + B) = −5665 check: 33² + 56² = 1089 + 3136 = 4225 = 65² ✓

WE 5

Prove a compound angle identity

Prove that sin(x + π/3) + sin(x − π/3) = sin x.

Step 1: Expand each term using sin(A ± B) sin(x + π/3) = sin x cos(π/3) + cos x sin(π/3) sin(x − π/3) = sin x cos(π/3) − cos x sin(π/3) Step 2: Add — the cos x sin(π/3) terms cancel LHS = 2 sin x cos(π/3) Step 3: Use cos(π/3) = 1/2 = 2 sin x · (1/2) = sin x = RHS ✓ proved

WE 6

Solve sin(x + 30°) = cos x

Solve the equation sin(x + 30°) = cos x for 0° ≤ x ≤ 360°.

Step 1: Expand the left side sin x cos 30° + cos x sin 30° = cos x (√3/2) sin x + (1/2) cos x = cos x Step 2: Move cos x terms to one side (√3/2) sin x = cos x − (1/2) cos x (√3/2) sin x = (1/2) cos x Step 3: Divide by cos x and rearrange tan x = (1/2) ÷ (√3/2) = 1/√3 Step 4: Solve in [0°, 360°] x = 30° or x = 30° + 180° = 210° x = 30°, 210° check x = 30°: sin 60° = √3/2 = cos 30° ✓

💡 Top tips

Watch the cos sign flip. cos(A + B) uses minus, cos(A − B) uses plus — the opposite of what students expect.
Choose the right split. For 15°, try 45° − 30°; for 105°, try 60° + 45°. Both work — pick the cleaner one.
For “given sin/cos” questions, use the Pythagorean identity to find the missing partner first, then plug into the compound formula.
Quadrant check: A acute → both sin and cos positive; B obtuse → sin positive, cos negative. The signs must reflect this.
Decimal sanity check: after getting an exact answer, evaluate it as a decimal and compare to the calculator value of the original angle.

⚠ Common mistakes

Treating sin(A + B) as sin A + sin B. It’s not — that’s the most common slip in the topic.
Wrong sign in the cos formula. cos(A + B) = cos A cos B − sin A sin B (the sign flips between LHS and RHS).
Forgetting to find missing values. Given sin A only, you must use sin² + cos² = 1 to get cos A before using a compound formula.
Sign errors from quadrant. If B is obtuse, cos B is negative — taking the positive square root gives the wrong answer.
Leaving surds unsimplified. Always rationalise denominators (e.g., 1/√3 → √3/3) and combine like terms.

Next note: Double Angle Formulae. Setting B = A in the compound angle formulae gives the double angle versions: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ, tan 2θ = 2 tan θ /(1 − tan²θ). Same logic, single-angle answer.

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