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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