Right Triangle Trigonometry

In a right triangle, the three basic ratios relate an angle to two sides.

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

Remembered as SOH-CAH-TOA.

"Opposite" and "adjacent" are relative to the angle you are working with, so they swap when you switch angles. The hypotenuse never changes.

To find a missing side, set up the ratio containing the two sides you care about and solve.

To find a missing angle, use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹. If sin θ = 0.6, then θ = sin⁻¹(0.6) ≈ 36.87°.

Check your calculator is in degrees, not radians. That single setting causes more wrong answers than any concept here.

The ratios are well defined because all right triangles with a given acute angle are similar, so corresponding side ratios are invariant — the same similarity argument that makes slope well defined.

The inverse functions sin⁻¹, cos⁻¹, tan⁻¹ return the unique angle within a restricted range ([−90°, 90°] for sin⁻¹ and tan⁻¹, [0°, 180°] for cos⁻¹), a restriction required because the trig functions are not one-to-one. The notation again denotes an inverse function rather than a reciprocal; the reciprocals are separately named cosecant, secant and cotangent.

The Pythagorean identity sin²θ + cos²θ = 1 is the Pythagorean theorem applied to a triangle with hypotenuse 1, and tan θ = sin θ / cos θ follows directly from the definitions.

Beyond right triangles, the Law of Sines a/sin A = b/sin B = c/sin C and the Law of Cosines c² = a² + b² − 2ab·cos C handle oblique triangles, with the Law of Cosines reducing to Pythagoras when C = 90°.