Trigonometry Cheat Sheet
Core trig identities, ratios, unit circle values, and laws of sines and cosines.
Reference
Right triangle ratios (angle θ)
| Function | Ratio |
|---|---|
| sin θ | opposite / hypotenuse |
| cos θ | adjacent / hypotenuse |
| tan θ | opposite / adjacent |
| csc θ | 1 / sin θ |
| sec θ | 1 / cos θ |
| cot θ | 1 / tan θ |
Unit circle (common angles)
| θ | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
| 120° | √3/2 | −1/2 | −√3 |
| 135° | √2/2 | −√2/2 | −1 |
| 150° | 1/2 | −√3/2 | −√3/3 |
| 180° | 0 | −1 | 0 |
| 270° | −1 | 0 | undefined |
| 360° | 0 | 1 | 0 |
Pythagorean identities
- sin²θ + cos²θ
- = 1
- 1 + tan²θ
- = sec²θ
- 1 + cot²θ
- = csc²θ
Double-angle
- sin 2θ
- = 2 sin θ · cos θ
- cos 2θ
- = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
- tan 2θ
- = 2 tan θ / (1 − tan²θ)
Sum / difference
- sin(α ± β)
- = sin α cos β ± cos α sin β
- cos(α ± β)
- = cos α cos β ∓ sin α sin β
- tan(α ± β)
- = (tan α ± tan β) / (1 ∓ tan α tan β)
Laws
- Law of sines
- a / sin A = b / sin B = c / sin C
- Law of cosines
- c² = a² + b² − 2ab cos C
Notes
- Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
- sin is odd: sin(−θ) = −sin θ. cos is even: cos(−θ) = cos θ.
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