Sin, Cos, Tan, Cot of an Angle
Exact value in square roots and the decimal, with reduction to an acute angle on the unit circle
Sin, Cos, Tan, Cot of an Angle
Values of Sine, Cosine, Tangent and Cotangent
On the unit circle the point of angle α is (cos α, sin α); tan α = sin α / cos α, cot α = cos α / sin α.
| Angle | sin | cos | tan | cot |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | — |
| 30° | 1/2 | √3/2 | √3/3 | √3 |
| 45° | √2/2 | √2/2 | 1 | 1 |
| 60° | √3/2 | 1/2 | √3 | √3/3 |
| 90° | 1 | 0 | — | 0 |
| 120° | √3/2 | −1/2 | −√3 | −√3/3 |
| 135° | √2/2 | −√2/2 | −1 | −1 |
| 150° | 1/2 | −√3/2 | −√3/3 | −√3 |
| 180° | 0 | −1 | 0 | — |
| 270° | −1 | 0 | — | 0 |
| 360° | 0 | 1 | 0 | — |
— — the value is undefined (division by zero)
Example: sin 30°
sin 30° = 1/2 = 0.5
P(cos 30°, sin 30°) = (√3/2, 1/2)
Half of an equilateral triangle with side 1: the hypotenuse is 1 and the leg opposite 30° is 1/2.
sin 30° = 1/2
P(cos 30°, sin 30°) ≈ (0.8660, 0.5)
Highlighted: the projection on the y-axis — sine, y = 0.5.
Common mistakes
- Radian mode on a calculator: sin 30 gives −0.9880, the sine of 30 radians. Switch it to degrees (DEG): sin 30° = 1/2.
- tan 90° and cot 0° do not exist: cos 90° = 0 and sin 0° = 0, and the ratio would divide by zero.
- The sign comes from the quadrant, not from the reference angle: sin 150° = 1/2, but sin 210° = −1/2 — both reduce to 30°.
See Also
- Arcsin, Arccos, Arctan, Arccot - The angle from the value of its sine, cosine, tangent or cotangent: exact in degrees and π, with the check and the unit circle
- Square Root Calculator - Calculate the square root of any number
- Simplify Square Root Calculator - Write any square root in simplest radical form, step by step
- Geometry Formulas: Area, Volume, Perimeter - The formula for every standard shape, where it comes from, and worked examples
FAQ
How to remember the table of values from 0° to 90°?
Write the sines of 0°, 30°, 45°, 60°, 90° as √0/2, √1/2, √2/2, √3/2, √4/2 — that is 0, 1/2, √2/2, √3/2, 1. Cosines are the same numbers in reverse order. Then tan = sin / cos: tan 30° = √3/3, tan 45° = 1, tan 60° = √3.
How to find the sine of an angle greater than 90°?
Subtract full turns of 360°, then reduce to the acute reference angle through 180° or 360° — the function stays the same, the sign comes from the quadrant: sin 150° = sin(180° − 30°) = sin 30°, so sin 150° = 1/2 and cos 150° = −√3/2. Past 360°: sin 570° = −1/2.
Why is tan 90° undefined?
tan = sin / cos, and cos 90° = 0: you can't divide by zero. On the unit circle line OP of 90° runs parallel to the tangent line x = 1 and never meets it. The same holds for 270°, 450° and every 90° + 180°·k.