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Pure · Trigonometry

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Trigonometric identities

The two AS identities — tan ≡ sin/cos and sin²+cos² ≡ 1 — where they come from, and how to use them to simplify expressions and prove "show that" results.

Pure · Trigonometry

Trigonometric identities

The two AS identities — tan ≡ sin/cos and sin²+cos² ≡ 1 — where they come from, and how to use them to simplify expressions and prove "show that" results.

Why it works

Two identities do almost all the work at AS, and they're both just facts you already know in disguise.

1. tan⁡θ≡sin⁡θcos⁡θ\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}. In a right-angled triangle tan⁡=oppadj\tan = \dfrac{\text{opp}}{\text{adj}}. Divide top and bottom by the hypotenuse and that's opp/hypadj/hyp=sin⁡θcos⁡θ\dfrac{\text{opp}/\text{hyp}}{\text{adj}/\text{hyp}} = \dfrac{\sin\theta}{\cos\theta}. It holds for every angle (wherever cos⁡θ≠0\cos\theta \ne 0).

2. sin⁡2θ+cos⁡2θ≡1\sin^2\theta + \cos^2\theta \equiv 1 (the Pythagorean identity). On a circle of radius 11, the point at angle θ\theta is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta), and it's distance 11 from the centre. Pythagoras on that: cos⁡2θ+sin⁡2θ=12\cos^2\theta + \sin^2\theta = 1^2. It is literally Pythagoras.

The "≡\equiv" sign means identically equal — true for every value of θ\theta. So you don't solve an identity; you use it (to simplify, or to prove another statement).

The rearrangements you'll reach for constantly: sin⁡2θ=1−cos⁡2θ,cos⁡2θ=1−sin⁡2θ.\sin^2\theta = 1 - \cos^2\theta, \qquad \cos^2\theta = 1 - \sin^2\theta. These let you swap sin⁡2\sin^2 for cos⁡2\cos^2 — the key move for turning a mixed expression into one written in a single function.

Simplifying. Replace tan⁡\tan by sin⁡cos⁡\frac{\sin}{\cos}, and hunt for sin⁡2+cos⁡2\sin^2 + \cos^2 to collapse to 11. For example 1−cos⁡2θsin⁡θ=sin⁡2θsin⁡θ=sin⁡θ\dfrac{1 - \cos^2\theta}{\sin\theta} = \dfrac{\sin^2\theta}{\sin\theta} = \sin\theta.

Proving "show that LHS ≡ RHS". Pick the messier side and work only that side until it turns into the other — never juggle both at once. Reliable tactics: convert everything to sin⁡\sin and cos⁡\cos; combine fractions over a common denominator; then look for sin⁡2+cos⁡2=1\sin^2 + \cos^2 = 1.

(One notation trap: sin⁡2θ\sin^2\theta means (sin⁡θ)2(\sin\theta)^2, not sin⁡(θ2)\sin(\theta^2) and not 2sin⁡θ2\sin\theta.)