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20 May 20252 min read

Every trig identity is hiding in one triangle and one circle

Students try to memorise a long list of trig identities and inevitably blank in the exam. Almost all of them fall out of two pictures you already know — so learn the pictures, not the list.

The trig identities chapter is where memorisation goes to die. Students write the identities on flashcards, recite them, and then sit in an exam unable to recall whether it's 1+tan⁡2θ1 + \tan^2\theta or 1−tan⁡2θ1 - \tan^2\theta. The list is long, the symbols are similar, and rote memory cracks under pressure precisely when you need it.

The good news: you barely need to memorise anything. Nearly every identity at this level is a consequence of two pictures, and if you can rebuild them you never have to recall them.

The first picture: a right-angled triangle

Two of the most-used identities are just definitions read off a triangle. Tangent is opposite over adjacent; sine is opposite over hypotenuse; cosine is adjacent over hypotenuse. Divide the first two and the hypotenuse cancels:

sin⁡θcos⁡θ=opp/hypadj/hyp=oppadj=tan⁡θ.\frac{\sin\theta}{\cos\theta} = \frac{\text{opp}/\text{hyp}}{\text{adj}/\text{hyp}} = \frac{\text{opp}}{\text{adj}} = \tan\theta.

You didn't memorise tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} — you derived it in one line from things you can't forget. That's the whole strategy.

The second picture: the unit circle and Pythagoras

The famous one, sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, isn't a fact to learn — it's the Pythagorean theorem in disguise. On the unit circle, a point has coordinates (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) and sits exactly one unit from the centre. Apply a2+b2=c2a^2 + b^2 = c^2 to that radius:

cos⁡2θ+sin⁡2θ=12=1.\cos^2\theta + \sin^2\theta = 1^2 = 1.

And the other two "identities" you were told to memorise are just this one, divided through. Divide every term by cos⁡2θ\cos^2\theta and you get tan⁡2θ+1=sec⁡2θ\tan^2\theta + 1 = \sec^2\theta. Divide by sin⁡2θ\sin^2\theta instead and you get 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta. Same identity, three outfits.

Don't memorise three Pythagorean identities. Memorise one, and know how to divide.

So the revision plan isn't "learn the list." It's: can you draw the triangle, can you draw the unit circle, and can you do the one or two lines of algebra that turn them into whatever the question needs? A student who can regenerate the identities walks into the exam with nothing to forget — which is a far calmer place to be than trusting a flashcard to survive the nerves.

#trigonometry#problem-solving#study-skills