1. Why Do We Need Inverse Trigonometric Functions?
Trigonometric functions like sin, cos and tan take an angle and give a ratio. But sometimes we know the ratio and need to find the angle. For this, we use inverse trigonometric functions.
Examples:
- If \( \sin \theta = \dfrac{1}{2} \), then \( \theta = \sin^{-1}\left(\dfrac{1}{2}\right) \).
- If \( \tan \theta = 1 \), then \( \theta = \tan^{-1}(1) \).
Thus, inverse functions let us “reverse” trigonometric functions.
2. Why We Restrict Domains Before Taking Inverses
Trigonometric functions are periodic, meaning they repeat their values. For example,
\( \sin \theta = \sin(\theta + 2\pi) \)
Because of this repetition, they are not one-to-one, so they don’t have inverses unless we restrict their domains.
To define an inverse formally, we choose an interval where each function is:
- One-to-one
- Continuous
- Covers all possible output values
3. Definitions of Inverse Trigonometric Functions
After restricting domains, we define the six inverse trigonometric functions as follows:
\( y = \sin^{-1}(x) \iff \sin y = x, \; y \in [-\dfrac{\pi}{2}, \dfrac{\pi}{2}] \)
\( y = \cos^{-1}(x) \iff \cos y = x, \; y \in [0, \pi] \)
\( y = \tan^{-1}(x) \iff \tan y = x, \; y \in (-\dfrac{\pi}{2}, \dfrac{\pi}{2}) \)
\( y = \cot^{-1}(x) \iff \cot y = x, \; y \in (0, \pi) \)
\( y = \sec^{-1}(x) \iff \sec y = x, \; y \in [0, \pi], \; y \neq \dfrac{\pi}{2} \)
\( y = \csc^{-1}(x) \iff \csc y = x, \; y \in [-\dfrac{\pi}{2}, \dfrac{\pi}{2}], \; y \neq 0 \)
4. Principal Value Branches
Each inverse function returns a unique output called the principal value. It lies in the restricted interval chosen for that function.
4.1. Example of Principal Value
\( \sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right) = \dfrac{\pi}{3} \)
Even though the sine of many angles equals \(\dfrac{\sqrt{3}}{2}\), the inverse returns the principal value within its chosen interval.
5. Examples of Using Inverse Trigonometric Definitions
Example 1: Solve \( \cos^{-1}(1) \).
\( \cos y = 1 \Rightarrow y = 0 \)
Example 2: Find \( \tan^{-1}(-1) \).
\( \tan y = -1 \Rightarrow y = -\dfrac{\pi}{4} \)
since principal values of tan⁻¹ lie in \((-\dfrac{\pi}{2}, \dfrac{\pi}{2})\).