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

Strategy for Trigonometric Equations

A trig equation can mix angles, ratios, and squares — too many tools to guess at random. This is the decision tree: ask three questions about the equation, and the right method falls out.

📘 What you need to know

Three questions: (1) Same angle? (2) One trig function? (3) Linear or quadratic?
Different angles (θ and 2θ) → use double angle formulas to match.
Function of θ inside (e.g., 3x − π/6) → substitute y = function, transform interval, solve for y, back-substitute.
More than one trig function → use sin² + cos² = 1, tan = sin/cos, or 1 + tan² = sec² to reduce to one.
Quadratic-looking → factor or use the quadratic formula; reject impossible roots (|sin|, |cos| > 1).
Reject domain failures — sec, cosec, cot, tan all have undefined points; check each candidate solution doesn’t blow up.
If totally stuck → rewrite everything in sin and cos. Brute-force, but it always reveals the structure.

The decision tree

Spot this	Move
Different multiples (2x and x, etc.)	double angle: sin 2x = 2 sin x cos x; cos 2x = 1 − 2 sin²x = 2 cos²x − 1
Function inside trig (ax + b)	substitute y = ax + b; transform interval; back-substitute
Both sin² and cos linearly (or vice versa)	swap sin² ↔ 1 − cos² (or cos² ↔ 1 − sin²) so all match the linear term
sec² with tan	1 + tan²x = sec²x → reduce to tan only
cosec² with cot	1 + cot²x = cosec²x → reduce to cot only
Several reciprocals + sin/cos	rewrite in sin and cos; clear fractions; factor
Common factor across terms	factor — never divide (you lose solutions)

🧭 Recipe — pick your method

Same angle? If not, use double or compound angle to match.
One trig function? If not, use a Pythagorean or quotient identity to reduce.
Move to one side, factor if there’s a common term.
Linear or quadratic? Solve accordingly; reject any value out of range.
Find all solutions in the interval, then check the original equation is defined for each.

Worked examples

WE 1

Different multiples → double angle

Solve sin 2x + sin x = 0 for 0° ≤ x ≤ 360°.

2x and x → use sin 2x = 2 sin x cos x 2 sin x cos x + sin x = 0 Factor (don’t divide!) sin x (2 cos x + 1) = 0 Solve each factor sin x = 0 → x = 0°, 180°, 360° cos x = −1/2 → x = 120°, 240° x = 0°, 120°, 180°, 240°, 360°

WE 2

Function inside → substitute

Solve 2 sin(3x − π/6) = 1 for 0 ≤ x ≤ π. Give exact answers.

Step 1: Substitute y = 3x − π/6 sin y = 1/2 Step 2: Transform interval (×3, then −π/6) −π/6 ≤ y ≤ 17π/6 Step 3: Solve sin y = 1/2 principal: y = π/6; second: π − π/6 = 5π/6 add 2π: y = 13π/6, 17π/6 (both in range) Step 4: Back-substitute x = (y + π/6)/3 y = π/6 → x = π/9 y = 5π/6 → x = π/3 y = 13π/6 → x = 7π/9 y = 17π/6 → x = π x = π9, π3, 7π9, π

WE 3

Two ratios → use Pythagorean

Solve 2 sin²x + 3 cos x = 0 for 0° ≤ x ≤ 360°.

Linear cos → swap sin² for 1 − cos²x 2(1 − cos²x) + 3 cos x = 0 2 − 2 cos²x + 3 cos x = 0 Quadratic in cos x 2 cos²x − 3 cos x − 2 = 0 (2 cos x + 1)(cos x − 2) = 0 Reject cos x = 2 (impossible) cos x = −1/2 → x = 120°, 240° x = 120°, 240°

WE 4

sec² with tan → use 1 + tan² = sec²

Solve 2 sec²x = 5 tan x for 0° ≤ x ≤ 360°. Give answers to 3 s.f.

sec² with tan → use 1 + tan²x = sec²x 2(1 + tan²x) = 5 tan x 2 tan²x − 5 tan x + 2 = 0 Factor (2 tan x − 1)(tan x − 2) = 0 tan x = 1/2 or tan x = 2 Solve each — tan repeats every 180° tan x = 1/2 → x ≈ 26.57°, 206.57° tan x = 2 → x ≈ 63.43°, 243.43° x ≈ 26.6°, 63.4°, 207°, 243° (3 s.f.)

WE 5

Combine — factor after double angle

Solve sin 2x = √3 cos x for 0° ≤ x ≤ 360°.

2x and x → use sin 2x = 2 sin x cos x 2 sin x cos x = √3 cos x Move to one side, factor (don’t divide by cos x!) 2 sin x cos x − √3 cos x = 0 cos x (2 sin x − √3) = 0 Solve each cos x = 0 → x = 90°, 270° sin x = √3/2 → x = 60°, 120° x = 60°, 90°, 120°, 270° dividing by cos x would have killed x = 90° and 270°

WE 6

Reciprocal trig — check domain at the end

Solve sec x + tan x = 2 cos x for 0 ≤ x ≤ 2π.

Step 1: Rewrite in sin and cos 1/cos x + sin x/cos x = 2 cos x (1 + sin x)/cos x = 2 cos x Step 2: Multiply both sides by cos x 1 + sin x = 2 cos²x = 2(1 − sin²x) Step 3: Quadratic in sin x 2 sin²x + sin x − 1 = 0 (2 sin x − 1)(sin x + 1) = 0 sin x = 1/2 or sin x = −1 Step 4: Find candidates and check domain (sec, tan need cos x ≠ 0) sin x = 1/2 → x = π/6, 5π/6 ✓ (cos ≠ 0) sin x = −1 → x = 3π/2 ✗ (cos x = 0, sec/tan undefined — REJECT) x = π6, 5π6 always re-check the original equation makes sense — multiplying by cos x can introduce phantom solutions

💡 Top tips

Same angle first, one ratio second. Always reduce both before trying to solve.
Match identity to the linear term. Linear cos x? Use sin² = 1 − cos². Linear sin x? Use cos² = 1 − sin².
Factor, never divide. Dividing by sin x or cos x kills the zero solutions.
Write in sin/cos when you’re stuck. It always reveals what’s going on.
Check candidates against the original equation, especially with reciprocals — sec, cosec, tan, cot have undefined points.

⚠ Common mistakes

Forgetting to transform the interval when substituting (× the coefficient first, then add/subtract).
Picking the wrong cos 2θ form. Match it to whatever’s linear in the equation.
Dividing instead of factoring. Loses solutions every time.
Accepting impossible roots like sin x = 2 or cos x = −3.
Phantom solutions from multiplying by cos x or sin x — always check candidates work in the original equation.

That closes Trigonometric Proof & Equation Strategies, and the trigonometry side of Topic 3. Next section: Vector Properties — vectors as arrows in 2D and 3D, magnitudes, unit vectors, the dot product, and angles between vectors.

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