What are inverse trigonometric functions?
Inverse trigonometric functions (also called arc functions) are the inverse operations of trigonometric functions. They find the angle when given a trigonometric ratio. arcsin (sin⁻¹) finds the angle whose sine is a given value, arccos (cos⁻¹) finds the angle whose cosine is a given value, and arctan (tan⁻¹) finds the angle whose tangent is a given value. These functions are essential in trigonometry, calculus, and many applications where you need to find angles from ratios.
Select the inverse function (arcsin, arccos, or arctan), enter the trigonometric ratio value, choose angle mode (degrees or radians), then click Get result to see the angle. No sign-up required. Use it to solve trigonometry problems, find angles, or understand inverse trig functions.
How to use this inverse trig calculator
Select the inverse function: arcsin (inverse sine), arccos (inverse cosine), or arctan (inverse tangent). Enter the trigonometric ratio value: for arcsin and arccos, enter values between -1 and 1 (since sine and cosine range from -1 to 1); for arctan, enter any real number. Choose angle mode: degrees (0-360°) or radians (0-2π). Click Get result to see the angle in your chosen mode. The calculator shows the result and explanation.
Understanding inverse trig functions
Inverse trig functions answer: "What angle has this trigonometric ratio?" For example, arcsin(0.5) asks "What angle has sine 0.5?" The answer is 30° (or π/6 radians). These functions are the inverse of regular trig functions: if sin(30°) = 0.5, then arcsin(0.5) = 30°. They're denoted as sin⁻¹, cos⁻¹, tan⁻¹ or arcsin, arccos, arctan. The range of outputs is restricted to principal values: arcsin and arctan return angles from -90° to 90° (-π/2 to π/2), arccos returns angles from 0° to 180° (0 to π).
Common inverse trig values
Common values: arcsin(0) = 0°, arcsin(0.5) = 30°, arcsin(√2/2) = 45°, arcsin(√3/2) = 60°, arcsin(1) = 90°. arccos(1) = 0°, arccos(√3/2) = 30°, arccos(√2/2) = 45°, arccos(0.5) = 60°, arccos(0) = 90°. arctan(0) = 0°, arctan(1) = 45°, arctan(√3) = 60°. These values correspond to special angles in trigonometry. Enter these values to verify and understand inverse trig relationships.
Applications of inverse trig functions
Inverse trig functions are used in navigation (finding bearings from coordinates), physics (calculating angles in projectile motion, forces), engineering (determining angles in structures, mechanisms), computer graphics (rotations, transformations), and solving trigonometric equations. They're essential when you know a ratio but need the angle, which is common in real-world problems involving triangles and circular motion.
Degrees vs radians
Angles can be measured in degrees (360° in a full circle) or radians (2π radians in a full circle). Degrees are more intuitive for everyday use, while radians are preferred in calculus and advanced mathematics. This calculator supports both modes. 1 radian = 180°/π ≈ 57.3°. Common conversions: 30° = π/6 rad, 45° = π/4 rad, 60° = π/3 rad, 90° = π/2 rad, 180° = π rad. Choose the mode appropriate for your context.
Tips for using inverse trig functions
Remember domain restrictions: arcsin and arccos only accept values from -1 to 1, while arctan accepts any real number. Understand principal values: inverse functions return angles in specific ranges to ensure they're functions (one output per input). For angles outside principal ranges, add or subtract multiples of 180° (or π radians). When solving equations, consider all possible angles, not just principal values. Use this calculator to check your work and understand inverse trig relationships.
Summary
This inverse trigonometric functions calculator computes arcsin, arccos, and arctan values. Select function, enter ratio value, choose angle mode, and click Get result to see the angle. It's free, works in your browser, and requires no account. Use it to solve trigonometry problems, find angles from ratios, or understand inverse trig functions. Inverse trig functions are essential in mathematics, physics, and engineering for finding angles when trigonometric ratios are known.