Inverse Trigonometric Functions Calculator
Enter a value and choose which inverse trigonometric function to evaluate. The calculator returns the angle in both degrees and radians, shows the step-by-step working, and highlights the principal value range and domain restrictions for the chosen function. All six functions are supported: arcsin, arccos, arctan, arccot, arcsec, and arccsc.
What are inverse trigonometric functions?
Inverse trigonometric functions answer the question "what angle produces this ratio?" When you know that sin(30 deg) = 0.5, the inverse asks: given 0.5, what angle has that sine? The answer is arcsin(0.5) = 30 degrees. Each of the six standard trig functions has an inverse: arcsin (or sin^-1), arccos (or cos^-1), arctan (or tan^-1), arccot, arcsec, and arccsc. Because trig functions are periodic and many-to-one, the inverse is defined only over a restricted interval called the principal value range, which ensures each input maps to exactly one output.
Domains, ranges, and principal values
Each inverse function has a specific domain (the set of allowed inputs) and a principal value range (the interval of outputs). arcsin and arccos accept x in [-1, 1] only, because sine and cosine are always between -1 and 1. arctan and arccot accept any real number, because tangent and cotangent cover all reals. arcsec and arccsc require |x| >= 1, because secant and cosecant are never between -1 and 1. The principal value ranges are: arcsin in [-90, 90] deg, arccos in [0, 180] deg, arctan in (-90, 90) deg, arccot in (0, 180) deg, arcsec in [0, 180] deg excluding 90 deg, arccsc in [-90, 90] deg excluding 0 deg. Knowing these ranges tells you which quadrant your angle sits in.
How the formulas work
arcsin, arccos, and arctan are built in to every scientific calculator and programming language. arccot(x) = pi/2 - arctan(x), because cotangent is the co-function of tangent. arcsec(x) = arccos(1/x), because secant is the reciprocal of cosine. arccsc(x) = arcsin(1/x), because cosecant is the reciprocal of sine. These reciprocal identities let you evaluate all six inverse functions using only arcsin, arccos, and arctan. To verify any result, apply the forward trig function to your answer and check you get x back: for example, sin(arcsin(0.5)) = 0.5.
Practical applications
Inverse trig functions appear across engineering, physics, navigation, and computer graphics. In physics, you use arctan to find the angle of a force vector from its components. In construction, you use arcsin to find a roof pitch angle from a rise-over-run ratio. In navigation, arccos appears in the formula for the great-circle distance between two points on a sphere. In game programming, arctan2 (a two-argument form) converts Cartesian coordinates to polar angles. The complementary identities - arcsin(x) + arccos(x) = 90 deg and arctan(x) + arccot(x) = 90 deg - are useful shortcuts that often simplify problems.
Common inverse trig values
| x | arcsin(x) | arccos(x) | arctan(x) |
|---|---|---|---|
| -1 | -90 deg = -pi/2 | 180 deg = pi | n/a |
| -sqrt(3)/2 (-0.866) | -60 deg = -pi/3 | 150 deg = 5pi/6 | n/a |
| -sqrt(2)/2 (-0.707) | -45 deg = -pi/4 | 135 deg = 3pi/4 | n/a |
| -1/2 (-0.5) | -30 deg = -pi/6 | 120 deg = 2pi/3 | n/a |
| 0 | 0 deg = 0 | 90 deg = pi/2 | 0 deg = 0 |
| 1/2 (0.5) | 30 deg = pi/6 | 60 deg = pi/3 | n/a |
| sqrt(2)/2 (0.707) | 45 deg = pi/4 | 45 deg = pi/4 | n/a |
| sqrt(3)/2 (0.866) | 60 deg = pi/3 | 30 deg = pi/6 | n/a |
| 1 | 90 deg = pi/2 | 0 deg = 0 | n/a |
| -sqrt(3) (-1.732) | n/a | n/a | -60 deg = -pi/3 |
| -1 | n/a | n/a | -45 deg = -pi/4 |
| -1/sqrt(3) (-0.577) | n/a | n/a | -30 deg = -pi/6 |
| 1/sqrt(3) (0.577) | n/a | n/a | 30 deg = pi/6 |
| 1 | n/a | n/a | 45 deg = pi/4 |
| sqrt(3) (1.732) | n/a | n/a | 60 deg = pi/3 |
Exact principal values for frequently used inputs. Memorising these is useful for exams and mental calculation.
Frequently asked questions
What is the difference between sin^-1 and 1/sin?
sin^-1(x) is notation for arcsin(x), the inverse sine function that returns an angle. It is NOT the same as 1/sin(x), which is cosecant (csc). The superscript -1 means "inverse function," not "reciprocal." This is a common source of confusion: in function notation, f^-1 always means the inverse function. The reciprocal of sin(x) is written as (sin(x))^-1 or 1/sin(x) = csc(x).
Why does arcsin only accept values between -1 and 1?
The sine function always outputs values in the range [-1, 1], no matter what angle you input. If you ask "what angle has a sine of 2?", there is no real answer because sine can never reach 2. The domain of arcsin is therefore restricted to [-1, 1], mirroring the output range of sine. Inputs outside this range are undefined in the real numbers (though they have complex-number representations).
How do I convert the result from radians to degrees?
Multiply the radian value by 180 and divide by pi: degrees = radians * (180 / pi). For example, pi/6 radians * (180/pi) = 30 degrees. This calculator shows both units simultaneously, so you do not need to convert manually. To go the other way, multiply degrees by pi/180 to get radians.
What is the principal value and why does it matter?
Because trig functions repeat, multiple angles can have the same sine, cosine, or tangent. For example, sin(30 deg) = sin(150 deg) = 0.5. To make arcsin a proper function (one output per input), mathematicians restrict the output to a single interval called the principal value range: for arcsin this is [-90, 90] degrees. The calculator always returns the principal value. If you need angles outside that range, use the full-circle solutions: the general solution for arcsin(x) is the principal value plus any integer multiple of 360 degrees, or 180 degrees minus the principal value plus any integer multiple of 360 degrees.
What is the relationship between arcsin and arccos?
arcsin(x) + arccos(x) = 90 degrees (or pi/2 radians) for any x in [-1, 1]. This is the co-function identity: since sine and cosine are each other's co-functions (cos(angle) = sin(90 - angle)), their inverses always add to 90 degrees. Similarly, arctan(x) + arccot(x) = 90 degrees for any real x.
How do I find the angle in a right triangle using inverse trig?
Label the sides: opposite (O), adjacent (A), and hypotenuse (H) relative to the angle you want. If you know O and H, the angle = arcsin(O/H). If you know A and H, the angle = arccos(A/H). If you know O and A, the angle = arctan(O/A). The ratio you use depends on which two sides you know. This is the most common real-world application of inverse trig functions.