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Lesson 25 · Choosing the ratio
The side nobody mentions is the tell
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The side nobody mentions is the tell

Three ratios, three pairs of sides. sin⁡θ=oh\sin\theta = \dfrac{o}{h} pairs the opposite with the hypotenuse, cos⁡θ=ah\cos\theta = \dfrac{a}{h} pairs the adjacent with the hypotenuse, and tan⁡θ=oa\tan\theta = \dfrac{o}{a} pairs the two legs and leaves the hypotenuse out of it entirely. SOH CAH TOA is not a spell; it is an index.

So the pick is mechanical, and it takes five seconds. Name the side you KNOW. Name the side you WANT. Two sides name exactly one ratio — and the third side, the one the story never mentions, is the tell. If nobody measured the slanted length, the hypotenuse is out and you are in tangent country. Do that naming before any number touches the calculator, and the arithmetic afterwards has nowhere left to go wrong.

One more habit worth building now: the hypotenuse is the longest side, so any answer that makes a leg outrun it is wrong before you check it. sin⁡θ\sin\theta and cos⁡θ\cos\theta never leave the range 00 to 11 for an acute angle — multiplying by them SHRINKS, dividing by them GROWS. That single sentence catches most of the wrong answers below.

θoh

sin⁡θ=oh\sin\theta = \frac{o}{h}

  • θ\theta= Acute angle (angle)
  • oo= Opposite side (length)
  • hh= Hypotenuse (length)
Right-Triangle Sine Ratio (SOH) solver →
θah

cos⁡θ=ah\cos\theta = \frac{a}{h}

  • θ\theta= Acute angle (angle)
  • aa= Adjacent side (length)
  • hh= Hypotenuse (length)
Right-Triangle Cosine Ratio (CAH) solver →
θoa

tan⁡θ=oa\tan\theta = \frac{o}{a}

  • θ\theta= Acute angle (angle)
  • oo= Opposite side (length)
  • aa= Adjacent side (length)
Right-Triangle Tangent Ratio (TOA) solver →