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
Arcsin, Arccos, Arctan, Arccot
Arcsine, Arccosine, Arctangent and Arccotangent
arcsin x is the angle in [−90°, 90°] whose sine is x; arccos x, in [0°, 180°]; arctan x, in (−90°, 90°); arccot x, in (0°, 180°).
| x | arcsin x | arccos x |
|---|---|---|
| −1 | −90° = −π/2 | 180° = π |
| −√3/2 | −60° = −π/3 | 150° = 5π/6 |
| −√2/2 | −45° = −π/4 | 135° = 3π/4 |
| −1/2 | −30° = −π/6 | 120° = 2π/3 |
| 0 | 0° = 0 | 90° = π/2 |
| 1/2 | 30° = π/6 | 60° = π/3 |
| √2/2 | 45° = π/4 | 45° = π/4 |
| √3/2 | 60° = π/3 | 30° = π/6 |
| 1 | 90° = π/2 | 0° = 0 |
| x | arctan x | arccot x |
|---|---|---|
| −√3 | −60° = −π/3 | 150° = 5π/6 |
| −1 | −45° = −π/4 | 135° = 3π/4 |
| −√3/3 | −30° = −π/6 | 120° = 2π/3 |
| 0 | 0° = 0 | 90° = π/2 |
| √3/3 | 30° = π/6 | 60° = π/3 |
| 1 | 45° = π/4 | 45° = π/4 |
| √3 | 60° = π/3 | 30° = π/6 |
Example: arctan(1)
arctan(1) = 45° = π/4
arctan(1) is the angle α in (−90°, 90°) with tan α = 1.
α = arctan(1) = 45°
P(cos 45°, sin 45°) = (√2/2, √2/2)
T(1, 1) is where line OP meets the line x = 1; highlighted: tan α = 1.
Common mistakes
- sin⁻¹ x means arcsin x, not 1/sin x: sin⁻¹(1/2) = 30°, while 1/sin 30° = 2.
- Calculator mode: in RAD, arcsin(1/2) gives 0.5236, which is radians; in degrees (DEG) arcsin(1/2) = 30°.
- arcsin and arccos exist only for −1 ≤ x ≤ 1: arcsin 2 does not exist, while arctan(2) ≈ 63.4349°.
See Also
- 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
- 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
FAQ
How to find arcsin from a table of sines?
Find in the table of sines the angle in [−90°, 90°] whose sine is x: sin 30° = 1/2, so arcsin(1/2) = 30°. For a negative x take the angle with a minus sign: arcsin(−1/2) = −30°.
What is the difference between arcsin, sin⁻¹ and 1/sin?
sin⁻¹ x is the same as arcsin x — that is how the calculator key is labelled. 1/sin x is a different function, the cosecant. arcsin x is the angle whose sine is x; arctan x likewise is written tan⁻¹ x.
Why does arcsin 2 not exist?
The sine is the y-coordinate of a point on the unit circle, so it lies between −1 and 1, and no angle has a sine of 2. The same holds for arccos: the cosine is the x-coordinate, also between −1 and 1. arctan and arccot exist for any x.