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Geometry & measures · Pythagoras & right-angled trigonometry

Chapter 1 · 4

The idea

Right-angled trigonometry (SOHCAHTOA)

Labelling opposite, adjacent and hypotenuse from the angle you are using, choosing the right ratio, finding sides and angles, the exact values that must be known by heart, and angles of elevation and depression.

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Geometry & measures · Pythagoras & right-angled trigonometry

Right-angled trigonometry (SOHCAHTOA)

Labelling opposite, adjacent and hypotenuse from the angle you are using, choosing the right ratio, finding sides and angles, the exact values that must be known by heart, and angles of elevation and depression.

Why it works

Fixed ratios with names

All triangles with the same angles are similar, so in a right-angled triangle the ratio of any two sides depends only on the angle — not on how big the triangle is. Those fixed ratios are named:

sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}}, \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}

remembered as SOH CAH TOA.

Label from the angle you are using

The hypotenuse never moves (opposite the right angle), but "opposite" and "adjacent" swap over depending on which angle you're working from. So: mark the angle, label the side facing it opp, the remaining short side adj, and only then choose the ratio containing the two sides you care about.

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