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How to Put Inverse Trig Functions in Calculator

Reviewed by Calculator Editorial Team

Inverse trigonometric functions allow you to find angles from known ratios. This guide explains how to use inverse trig functions on calculators, including step-by-step instructions, common functions, and practical examples.

How to Use Inverse Trig Functions

Inverse trigonometric functions (also called arcus functions) reverse the standard trigonometric functions. While sin(θ) gives you a ratio from an angle, inverse functions like arcsin(x) give you an angle from a ratio.

Basic Steps

  1. Identify the trigonometric function you need to invert (sine, cosine, or tangent)
  2. Enter the ratio value (between -1 and 1 for sine and cosine, any real number for tangent)
  3. Select the inverse function on your calculator
  4. Calculate and interpret the resulting angle in degrees or radians

Important Notes

Most calculators provide inverse trig functions in both degrees and radians. Always check your calculator's mode setting before performing calculations.

Calculator Methods

Modern scientific calculators typically have inverse trig functions accessible through the following methods:

Scientific Calculator

  1. Press the "2nd" or "SHIFT" function key
  2. Locate the trig function button (SIN, COS, TAN)
  3. Enter your ratio value
  4. Press the equals (=) key

Graphing Calculator

  1. Access the "Math" or "Trig" menu
  2. Select the inverse function (arcsin, arccos, arctan)
  3. Enter your ratio value in parentheses
  4. Execute the calculation

Formula Used

For a given ratio x, the inverse trigonometric functions are calculated as:

  • arcsin(x) = θ where sin(θ) = x
  • arccos(x) = θ where cos(θ) = x
  • arctan(x) = θ where tan(θ) = x

Common Inverse Trig Functions

There are three primary inverse trigonometric functions:

arcsin(x) - Inverse Sine

Finds the angle whose sine is x. Range: -π/2 to π/2 radians (-90° to 90°).

arccos(x) - Inverse Cosine

Finds the angle whose cosine is x. Range: 0 to π radians (0° to 180°).

arctan(x) - Inverse Tangent

Finds the angle whose tangent is x. Range: -π/2 to π/2 radians (-90° to 90°).

Range Considerations

The ranges for inverse trig functions are important because they determine the quadrant of the resulting angle. For example, arcsin(x) always returns an angle in the first or fourth quadrant.

Worked Examples

Example 1: arcsin(0.5)

To find the angle whose sine is 0.5:

  1. Set your calculator to degree mode
  2. Press 2nd then SIN
  3. Enter 0.5
  4. Press = to get 30°

Example 2: arccos(-0.866)

To find the angle whose cosine is -0.866:

  1. Set your calculator to radian mode
  2. Press 2nd then COS
  3. Enter -0.866
  4. Press = to get approximately 2.356 radians (135°)

FAQ

What is the difference between sin and arcsin?

The sine function (sin) takes an angle and returns a ratio. The inverse sine function (arcsin) takes a ratio and returns an angle. They are mathematical inverses of each other.

Why do inverse trig functions have range restrictions?

The range restrictions ensure that inverse trig functions return the principal value (the primary angle solution) within the function's domain. This makes the functions one-to-one and invertible.

Can I use inverse trig functions for any real number?

No. Inverse sine and cosine functions are only defined for inputs between -1 and 1. Inverse tangent accepts any real number input.