Interactively explore 30-60-90 and 45-45-90 special right triangles. Enter any one side to instantly find all three sides as exact expressions (with √2 and √3) and all six trig ratios as exact fractions.
| Angle | sin | cos | tan |
|---|
Ratios: x : x√3 : 2x
Derive it from an equilateral triangle. Start with an equilateral triangle of side 2. All angles are 60°. Bisect it straight down the middle — you get two congruent right triangles.
Each right triangle has: hypotenuse = 2, short leg = 1 (half the base), and long leg = √(2² − 1²) = √3. The three angles become 30°, 60°, 90°.
Scaling by x gives the general ratios: short = x, long = x√3, hyp = 2x.
Ratios: x : x : x√2
Derive it from a unit square. Draw a square with side 1 and slice it diagonally. You get two congruent right triangles, each with legs = 1 and hypotenuse = √(1² + 1²) = √2.
Both acute angles are 45° (isosceles right triangle). Scaling by x: leg = x, leg = x, hyp = x√2.
Unit circle connection: At 45° the point is (√2/2, √2/2), which is exactly cos 45° = sin 45° = 1/√2 = √2/2 — the same ratio as x : x√2 = 1 : √2.
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