Special angles: 30°, 45°, 60°
What you'll learn
Derive the exact trig values for 30°, 45°, and 60° from two triangles you can sketch from memory.
Two right triangles appear in almost every trig problem. Their side ratios come from pure geometry — no calculator needed — so their trig values are exact.
The 30-60-90 triangle
Start with an equilateral triangle (all sides = 2, all angles = 60°). Cut it in half down the middle. Each half is a right triangle with angles 30°-60°-90° and sides:
- Short leg (opposite 30°): 1
- Long leg (opposite 60°): √3
- Hypotenuse: 2
The 45-45-90 triangle
Draw the diagonal of a unit square (side = 1). The diagonal cuts it into two right triangles with angles 45°-45°-90° and sides:
- Two equal legs: 1
- Hypotenuse (Pythagorean theorem): √2
The exact values
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
These don't need memorizing if you remember the two triangles. Read the ratio straight off the sides: sin 30° = opposite/hyp = 1/2. Sin 60° uses the same triangle but from the other angle: now the opposite side is √3, so sin 60° = √3/2.
Why these two matter
Every angle in a standard trig table can be built from 30°, 45°, 60°, and their reflections and sums. Memorize the two triangles and you carry the core of trig in your head.
Check your understanding
Question 1 of 2
What is sin 45°?