Inverse Trigonometric Functions Calculator
Calculate inverse trigonometric functions instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Formula
arcsin(x), arccos(x), arctan(x), arccot(x), arcsec(x), arccsc(x)
Inverse trigonometric functions return the angle whose trigonometric ratio equals the input value. Each has a specific domain restriction and returns values in a principal range to ensure unique results.
Worked Examples
Example 1: Inverse Trig of 0.5
Problem: Find all six inverse trigonometric function values for input 0.5.
Solution: arcsin(0.5) = 30 degrees (0.523599 rad) - verified: sin(30) = 0.5\narccos(0.5) = 60 degrees (1.047198 rad) - verified: cos(60) = 0.5\narctan(0.5) = 26.565051 degrees (0.463648 rad) - verified: tan(26.565) = 0.5\narccot(0.5) = 63.434949 degrees (1.107149 rad)\narcsec(0.5) = undefined (|0.5| < 1)\narccsc(0.5) = undefined (|0.5| < 1)
Result: arcsin = 30 deg | arccos = 60 deg | arctan = 26.565 deg
Example 2: Inverse Trig of -1
Problem: Calculate all valid inverse trig values for input -1.
Solution: arcsin(-1) = -90 degrees (-pi/2 rad)\narccos(-1) = 180 degrees (pi rad)\narctan(-1) = -45 degrees (-pi/4 rad)\narccot(-1) = 135 degrees (3pi/4 rad)\narcsec(-1) = 180 degrees (pi rad)\narccsc(-1) = -90 degrees (-pi/2 rad)\nNote: arcsin(-x) = -arcsin(x) and arccos(-x) = 180 - arccos(x)
Result: arcsin = -90 deg | arccos = 180 deg | arctan = -45 deg
Frequently Asked Questions
What are inverse trigonometric functions?
Inverse trigonometric functions reverse the operation of the standard trigonometric functions. While sin, cos, and tan take an angle and return a ratio, the inverse functions (arcsin, arccos, arctan) take a ratio and return an angle. For example, if sin(30) = 0.5, then arcsin(0.5) = 30 degrees. These functions are written as arcsin(x), arccos(x), arctan(x), or equivalently as sin-1(x), cos-1(x), tan-1(x). Note that sin-1(x) does NOT mean 1/sin(x); it is a completely different function. Inverse trig functions are essential for finding unknown angles in triangles, physics problems, engineering calculations, and many other applications where the ratio of sides is known but the angle is needed.
What are the domains and ranges of inverse trig functions?
Each inverse trig function has a restricted domain and range to ensure it returns a unique value. Arcsin(x) has domain [-1, 1] and range [-90, 90] degrees ([-pi/2, pi/2] radians). Arccos(x) has domain [-1, 1] and range [0, 180] degrees ([0, pi] radians). Arctan(x) accepts all real numbers and has range (-90, 90) degrees ((-pi/2, pi/2) radians). Arccot(x) accepts all real numbers with range (0, 180) degrees ((0, pi) radians). Arcsec(x) requires |x| >= 1 with range [0, 180] excluding 90 degrees. Arccsc(x) requires |x| >= 1 with range [-90, 90] excluding 0 degrees. These ranges are called principal value branches and represent the standard conventions used in mathematics.
Why do inverse trig functions need restricted ranges?
Inverse trig functions need restricted ranges because the original trig functions are periodic and many-to-one: multiple angles produce the same ratio value. For example, sin(30) = sin(150) = 0.5, and infinitely many other angles also have sine equal to 0.5 (like 30 + 360, 150 + 360, etc.). For arcsin(0.5) to return a single definite answer, we must restrict the output to one interval. The convention chooses ranges that include the most commonly used angles: arcsin uses [-90, 90] to cover one complete period of increasing sine values, arccos uses [0, 180] for one complete period of decreasing cosine values, and arctan uses (-90, 90) for one complete period of increasing tangent values. Without these restrictions, the inverse functions would not be true functions in the mathematical sense.
What are the derivatives of inverse trig functions?
The derivatives of inverse trig functions are important results in calculus. The derivative of arcsin(x) is 1/sqrt(1-x2), valid for |x| < 1. The derivative of arccos(x) is -1/sqrt(1-x2), which is simply the negative of arcsin's derivative. The derivative of arctan(x) is 1/(1+x2), valid for all real x. The derivative of arccot(x) is -1/(1+x2). The derivative of arcsec(x) is 1/(|x|sqrt(x2-1)), and arccsc(x) has derivative -1/(|x|sqrt(x2-1)). These formulas are derived using implicit differentiation: if y = arcsin(x), then x = sin(y), differentiating gives 1 = cos(y) dy/dx, so dy/dx = 1/cos(y) = 1/sqrt(1-sin2(y)) = 1/sqrt(1-x2). These derivatives appear frequently as integration results.
How do you evaluate inverse trig functions of common values?
For common values, memorize the standard angle results. For arcsin: arcsin(0) = 0, arcsin(1/2) = 30, arcsin(sqrt(2)/2) = 45, arcsin(sqrt(3)/2) = 60, arcsin(1) = 90 degrees. For arccos: arccos(1) = 0, arccos(sqrt(3)/2) = 30, arccos(sqrt(2)/2) = 45, arccos(1/2) = 60, arccos(0) = 90 degrees. For arctan: arctan(0) = 0, arctan(1/sqrt(3)) = 30, arctan(1) = 45, arctan(sqrt(3)) = 60 degrees. Negative inputs flip the sign for arcsin and arctan (odd functions), while for arccos you subtract from 180 degrees: arccos(-x) = 180 - arccos(x). These values come directly from the well-known 30-60-90 and 45-45-90 triangle ratios.
How are inverse trig functions used in integration?
Inverse trig functions appear as results of many standard integrals in calculus. The integral of 1/sqrt(1-x2) dx is arcsin(x) + C. The integral of 1/(1+x2) dx is arctan(x) + C. The integral of 1/(x sqrt(x2-1)) dx is arcsec(|x|) + C. More generally, the integral of 1/sqrt(a2-x2) dx is arcsin(x/a) + C, and the integral of 1/(a2+x2) dx is (1/a)arctan(x/a) + C. These patterns are recognized through trigonometric substitution: when you see sqrt(1-x2), substitute x = sin(theta); when you see 1+x2, substitute x = tan(theta). Recognizing these integral forms is a fundamental skill in calculus and appears extensively in physics, engineering, and probability theory.