Trigonometry
Trigonometry studies the relationships between the sides and angles of triangles. It is essential for physics, engineering, architecture, and computer graphics.
Right Triangle Basics
In a right triangle, the sides have special names relative to an angle θ:
- Hypotenuse — the longest side, opposite the right angle
- Opposite — the side directly across from angle θ
- Adjacent — the side next to angle θ (not the hypotenuse)
The Pythagorean Theorem connects them: a² + b² = c² (where c is the hypotenuse).
SOH-CAH-TOA
The three main trig ratios let you find missing sides or angles:
The Ratios
sin θ = Opposite / Hypotenuse (SOH)
cos θ = Adjacent / Hypotenuse (CAH)
tan θ = Opposite / Adjacent (TOA)
Example
Right triangle: hypotenuse = 10, angle θ = 30°. Find the opposite side.
sin 30° = 0.5 = opposite/10
opposite = 10 × 0.5 = 5
Common Angle Values
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Inverse Trig Functions
If you know a ratio and want to find the angle, use inverse trig: sin⁻¹, cos⁻¹, tan⁻¹ (also written arcsin, arccos, arctan).
Finding an angle
If sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°
If tan θ = 1, then θ = tan⁻¹(1) = 45°
Quick Quiz
Test what you just learned. Choose the best answer for each question.