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Pure · Trigonometry

Chapter 1 · 4

The idea

Solving trigonometric equations

The full method for solving trig equations in a range — quadratics in sin/cos, using sin²+cos²≡1 to convert a mixed equation, and transformed arguments like sin 2x and cos(x − 30°) where the interval must change.

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Pure · Trigonometry

Solving trigonometric equations

The full method for solving trig equations in a range — quadratics in sin/cos, using sin²+cos²≡1 to convert a mixed equation, and transformed arguments like sin 2x and cos(x − 30°) where the interval must change.

Why it works

One skeleton

Trig equations look endless — quadratics in sin⁡\sin, mixed sin⁡\sin/cos⁡\cos, doubled angles — but the exam only ever dresses up ONE task. Every trig equation has the same skeleton: get it into the form sin⁡(something)=k\sin(\text{something}) = k (or cos⁡\cos, or tan⁡\tan), find the first value, then use the graph's symmetry to collect all solutions in the range. Three things make them harder — each with a standard fix.

Quadratics in a trig function

An equation like 2sin⁡2x−sin⁡x−1=02\sin^2x - \sin x - 1 = 0 is just a quadratic wearing a disguise:

2sin⁡2x−sin⁡x−1=0  →  s=sin⁡x    (2s+1)(s−1)=02\sin^2 x - \sin x - 1 = 0 \;\xrightarrow{\;s = \sin x\;}\; (2s + 1)(s - 1) = 0

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The rest of the explanation, plus 2 worked examples you step through move by move.

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