Understanding Inverse Trigonometric Functions

What are Inverse Trigonometric Functions?

Inverse trigonometric functions, also known as arcus functions or anti-trigonometric functions, are the inverse functions of the basic trigonometric functions. They are used to find the angle when given a trigonometric ratio. The three main inverse trigonometric functions are:

• Arcsine (arcsin or sin⁻¹)
• Arccosine (arccos or cos⁻¹)
• Arctangent (arctan or tan⁻¹)

Formulas and Definitions

For a right-angled triangle with an angle θ:

• \(\arcsin(x) = \theta\) where \(\sin(\theta) = x\) and \(-1 \leq x \leq 1\)
• \(\arccos(x) = \theta\) where \(\cos(\theta) = x\) and \(-1 \leq x \leq 1\)
• \(\arctan(x) = \theta\) where \(\tan(\theta) = x\) for all real x

Calculation Steps

1. Identify the inverse trigonometric function to use (arcsin, arccos, or arctan).
2. Input the value (between -1 and 1 for arcsin and arccos, any real number for arctan).
3. Apply the inverse trigonometric function.
4. If needed, convert the result from radians to degrees using the formula: angle in degrees = angle in radians × 180°/π.
5. Round the result to the desired number of decimal places.

Example Calculation

Let's calculate arcsin(0.5):

1. We use the arcsine function: arcsin(0.5)
2. Using a calculator or computer: arcsin(0.5) ≈ 0.5236 radians
3. Converting to degrees: 0.5236 × 180°/π ≈ 30°

Visual Representation

This diagram illustrates arcsin(0.5) on the unit circle. The red line represents the angle (30°), and the yellow arc shows the measure of the angle. The green dashed lines form a right triangle where the sine of the angle (opposite / hypotenuse) is 0.5.