Inverse Trigonometric Functions

Overview

Inverse trigonometric functions, such as arcsine, arccosine, and arctangent, allow us to find the angle corresponding to a given trigonometric value. These functions are useful in determining the angle from a known ratio, such as \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \).

Key Concepts

  • Inverse Sine (Arcsine): \( \sin^{-1}(x) \) gives the angle whose sine is \( x \), with a range of \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) or \( [-90^\circ, 90^\circ] \).
  • Inverse Cosine (Arccosine): \( \cos^{-1}(x) \) gives the angle whose cosine is \( x \), with a range of \( [0, \pi] \) or \( [0^\circ, 180^\circ] \).
  • Inverse Tangent (Arctangent): \( \tan^{-1}(x) \) gives the angle whose tangent is \( x \), with a range of \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) or \( (-90^\circ, 90^\circ) \).
  • Ranges and Domain: Each inverse trigonometric function has a restricted range to ensure the function is one-to-one and thus invertible.
  • Interpretation: The results from the inverse functions should be interpreted based on the unit circle and the ranges of the functions.

Calculating Angles Using Inverse Functions

To calculate an angle from a trigonometric ratio, use the appropriate inverse function. Here's how:

  • For \( \sin^{-1}(x) \): Find the angle where the sine is \( x \) within the range \( [-90^\circ, 90^\circ] \) or \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
  • For \( \cos^{-1}(x) \): Find the angle where the cosine is \( x \) within the range \( [0^\circ, 180^\circ] \) or \( [0, \pi] \).
  • For \( \tan^{-1}(x) \): Find the angle where the tangent is \( x \) within the range \( (-90^\circ, 90^\circ) \) or \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

Practice Problems

  1. Find \( \sin^{-1}(0.5) \) in degrees.
    Solution

    \( \sin^{-1}(0.5) = 30^\circ \). This is the angle whose sine is \( 0.5 \), and it falls within the range \( [-90^\circ, 90^\circ] \).

  2. Find \( \cos^{-1}(0.707) \) in degrees.
    Solution

    \( \cos^{-1}(0.707) = 45^\circ \). This is the angle whose cosine is \( 0.707 \), and it falls within the range \( [0^\circ, 180^\circ] \).

  3. Find \( \tan^{-1}(1) \) in degrees.
    Solution

    \( \tan^{-1}(1) = 45^\circ \). This is the angle whose tangent is \( 1 \), and it falls within the range \( (-90^\circ, 90^\circ) \).

  4. Find \( \sin^{-1}(-\frac{\sqrt{2}}{2}) \) in radians.
    Solution

    \( \sin^{-1}(-\frac{\sqrt{2}}{2}) = -\frac{\pi}{4} \) radians. This is the angle whose sine is \( -\frac{\sqrt{2}}{2} \), and it falls within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

  5. Find \( \cos^{-1}(-1) \) in radians.
    Solution

    \( \cos^{-1}(-1) = \pi \) radians. This is the angle whose cosine is \( -1 \), and it falls within the range \( [0, \pi] \).

  6. Find \( \tan^{-1}(-\sqrt{3}) \) in radians.
    Solution

    \( \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \) radians. This is the angle whose tangent is \( -\sqrt{3} \), and it falls within the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

  7. Find \( \sin^{-1}(0) \) in radians.
    Solution

    \( \sin^{-1}(0) = 0 \) radians. The sine of \( 0 \) is \( 0 \), and the angle lies at \( 0 \) radians within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

  8. Find \( \cos^{-1}(-\frac{1}{2}) \) in degrees.
    Solution

    \( \cos^{-1}(-\frac{1}{2}) = 120^\circ \). This is the angle whose cosine is \( -\frac{1}{2} \), and it falls within the range \( [0^\circ, 180^\circ] \).

  9. Find \( \tan^{-1}(\frac{1}{\sqrt{3}}) \) in degrees.
    Solution

    \( \tan^{-1}(\frac{1}{\sqrt{3}}) = 30^\circ \). This is the angle whose tangent is \( \frac{1}{\sqrt{3}} \), and it falls within the range \( (-90^\circ, 90^\circ) \).

→Law of Sines and Law of Cosines