Trigonometry A — Triangles & the Unit Circle

Right triangles — SOH-CAH-TOA

ELI5: in a right triangle, each acute angle is fixed by the ratio of two sides — not their size. Shrink the triangle and the ratios (hence the angle) stay put. The memory hook is SOH-CAH-TOA:

Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent

and the three sides obey the Pythagorean theorem a² + b² = c².

leg_{a} = 3

adjacent leg

leg_{b} = 4

opposite leg

hyp_{c} = \sqrt{leg_{a}^{2} + leg_{b}^{2}}

5

Pythagoras: the hypotenuse

\theta = \mathrm{atan}\left(\frac{leg_{b}}{leg_{a}}\right)

0.9273

angle from the tangent ratio (radians)

\theta_{deg} = \frac{\theta \cdot 180}{\pi}

53.1301

the same angle in degrees

right(leg_a, leg_b)

drawn to scale — note the right-angle mark

✓ pass

\mathrm{abs}\left(hyp_{c} - 5\right) < 10^{-9}

the famous 3-4-5 right triangle

Real-world hook: SOH-CAH-TOA is how a wheelchair ramp's angle is set from its rise and run, how surveyors turn a measured angle into a height, and how a game converts a joystick angle into x/y movement.

Try it yourself: a right triangle has legs 5 and 12. How long is the hypotenuse?

Show the answer

√(5² + 12²) = √169 = 13 (another classic triple).

hyp_{you} = 13

✏️ Your turn: find the hypotenuse of a 5-12 right triangle. The check confirms hyp² = 5² + 12² — so it never just states the answer.

✓ pass

\mathrm{abs}\left(hyp_{you}^{2} - 5^{2} + 12^{2}\right) < 10^{-9}

green when your hypotenuse satisfies Pythagoras