Find the angle whose sine, cosine, tangent, secant, cosecant or cotangent equals your value. Type a decimal, a fraction or a root like √3/2 and get the principal value in degrees and radians, the exact answer for special values, all six inverse functions at once and every matching angle from 0° to 360°.
Last updated:
Inverse function
Decimal, fraction or root: 0.5, -1/2, √2/2, sqrt(3), 2/√3
Enter your values to see the result.
Advertisement
Advertisement
How to use the Inverse Trig Calculator
Pick the inverse function: arcsin, arccos, arctan, arcsec, arccsc or arccot.
Type the value x as a decimal (0.5), a fraction (-1/2) or a root (√3/2 or sqrt(3)/2).
Choose degrees or radians for the main answer and press Calculate.
Read the principal value, with the exact form when x is a special value.
Use the tables to see all six inverse functions of x and the standard exact values.
Inverse trig functions: domain and range
Function
Also written
Domain (x)
Principal range
Identity
arcsin x
sin⁻¹ x
−1 ≤ x ≤ 1
−90° to 90°
arcsin(−x) = −arcsin x
arccos x
cos⁻¹ x
−1 ≤ x ≤ 1
0° to 180°
arccos x = 90° − arcsin x
arctan x
tan⁻¹ x
all real x
−90° to 90° (open)
arctan(−x) = −arctan x
arcsec x
sec⁻¹ x
|x| ≥ 1
0° to 180°, not 90°
arcsec x = arccos(1/x)
arccsc x
csc⁻¹ x
|x| ≥ 1
−90° to 90°, not 0°
arccsc x = arcsin(1/x)
arccot x
cot⁻¹ x
all real x
0° to 180° (open)
arccot x = 90° − arctan x
Exact values to remember
x
arcsin x
arccos x
x
arctan x
0
0
π/2 (90°)
0
0
1/2
π/6 (30°)
π/3 (60°)
√3/3
π/6 (30°)
√2/2
π/4 (45°)
π/4 (45°)
1
π/4 (45°)
√3/2
π/3 (60°)
π/6 (30°)
√3
π/3 (60°)
1
π/2 (90°)
0
2 + √3
5π/12 (75°)
−1/2
−π/6 (−30°)
2π/3 (120°)
−1
−π/4 (−45°)
Tips
Inverse is not reciprocal. sin⁻¹(0.5) = 30°, but (sin 0.5)⁻¹ = 1 / sin 0.5 = csc 0.5.
Check your calculator mode. A handheld calculator in radian mode returns 0.5236 for sin⁻¹(0.5); in degree mode it returns 30.
Need the angle in a specific quadrant? Start from the principal value and use the "all angles in [0°, 360°)" line in the working.
For a right triangle, an acute angle is arctan(opposite/adjacent), arcsin(opposite/hypotenuse) or arccos(adjacent/hypotenuse).
Frequently asked questions
What is an inverse trig function?
An inverse trig function returns the angle that produces a given ratio. arcsin(x) is the angle whose sine is x, arccos(x) the angle whose cosine is x, and so on. They are also written sin⁻¹, cos⁻¹ and tan⁻¹, where the −1 means inverse, not a reciprocal.
Why does arcsin only give one angle?
Sine repeats, so infinitely many angles share the same sine. To make arcsin a function, its output is restricted to the principal range −90° to 90° (−π/2 to π/2). The calculator also lists every solution between 0° and 360° and the general solution with + 360°k.
What are the ranges of the inverse trig functions?
arcsin: [−π/2, π/2]; arccos: [0, π]; arctan: (−π/2, π/2); arcsec: [0, π] excluding π/2; arccsc: [−π/2, π/2] excluding 0; arccot: (0, π). These are the most common textbook conventions.
Why is arcsin(2) undefined?
The sine of any real angle lies between −1 and 1, so no angle has a sine of 2. arcsin and arccos accept only −1 ≤ x ≤ 1, while arcsec and arccsc accept only |x| ≥ 1 because secant and cosecant are never between −1 and 1. arctan and arccot accept any real number.
How do I calculate arcsec, arccsc and arccot?
Use the reciprocal identities: arcsec(x) = arccos(1/x), arccsc(x) = arcsin(1/x) and arccot(x) = arctan(1/x) for x > 0. For negative x, arccot(x) = π + arctan(1/x) so the answer stays in (0, π), and arccot(0) = π/2. Some software uses the range (−π/2, π/2] for arccot instead, which gives a different answer for negative x.
How do I convert the answer between degrees and radians?
Multiply radians by 180/π (about 57.2958) to get degrees, and multiply degrees by π/180 to get radians. For example, arctan(1) = π/4 rad = 45°.
Which values give exact answers?
Values built from 0, 1/2, √2/2, √3/2, 1, √3/3, √3, √2, 2 and 2√3/3 (and their negatives) give angles that are multiples of 30° or 45°, such as arccos(−1/2) = 2π/3 = 120°. The calculator also recognizes 2 ± √3 (15° and 75°) and (√5 ± 1)/4 (18° and 36° family).
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.