Inverse Trigonometric Functions: Definitions, Properties, and Applications

Inverse Trigonometric Functions

Introduction to Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric value. Since the six basic trigonometric functions are periodic and not one-to-one over their entire domains, we restrict their domains to intervals where they are one-to-one to define their inverses. These functions are essential for solving trigonometric equations and for applications in geometry and calculus.

• Key Point: The inverse trigonometric functions are defined by restricting the domains of the original trigonometric functions to intervals where they are one-to-one.

• Key Point: The principal values (ranges) of the inverse functions are chosen to ensure each input yields a unique output.

Inverse Sine Function (Arcsine)

Definition and Properties

The inverse sine function, also called arcsine, is denoted as $y = \arcsin(x)$ or $y = \sin^{-1}(x)$. It gives the unique angle $y$ in the interval $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ such that $\sin(y) = x$.

• Domain: $-1 \leq x \leq 1$

• Range: $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$

• Equation: $y = \arcsin(x) \iff \sin(y) = x$ and $y \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$

Evaluating Inverse Sine Without a Calculator

• Key Point: To find $\arcsin(x)$ for special values, locate the angle in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ whose sine is $x$.

• Example: $\arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$, since $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$.

Evaluating Inverse Sine With a Calculator

• Key Point: Use the calculator's inverse sine function to find approximate values in radians or degrees.

• Example: $\arcsin(-0.46) \approx -0.478$ radians.

Inverse Cosine Function (Arccosine)

Definition and Properties

The inverse cosine function, or arccosine, is denoted as $y = \arccos(x)$ or $y = \cos^{-1}(x)$. It gives the unique angle $y$ in the interval $[0, \pi]$ such that $\cos(y) = x$.

• Domain: $-1 \leq x \leq 1$

• Range: $[0, \pi]$

• Equation: $y = \arccos(x) \iff \cos(y) = x$ and $y \in [0, \pi]$

Inverse Tangent Function (Arctangent)

Definition and Properties

The inverse tangent function, or arctangent, is denoted as $y = \arctan(x)$ or $y = \tan^{-1}(x)$. It gives the unique angle $y$ in the interval $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$ such that $\tan(y) = x$.

• Domain: $-\infty < x < \infty$

• Range: $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$

• Equation: $y = \arctan(x) \iff \tan(y) = x$ and $y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$

End Behavior of Inverse Tangent Function

• Key Point: As $x \to \infty$, $\arctan(x) \to \frac{\pi}{2}$; as $x \to -\infty$, $\arctan(x) \to -\frac{\pi}{2}$.

Applications of Inverse Trigonometric Functions

Example: Calculating a Viewing Angle

Inverse trigonometric functions are used in real-world applications such as determining angles of elevation or depression. For example, to find the angle at which a photographer must look to see a rising balloon, use the arctangent function.

• Key Point: If the horizontal distance is 500 ft and the balloon's height is $s$, then $\tan\theta=\frac{s}{500}$implies that $\theta = \arctan\left(\frac{s}{500}\right)$.

• Key Point: The rate of change of $\theta$ with respect to $s$ is greater when "s" is small, due to the shape of the arctangent curve.

Summary Table: Inverse Trigonometric Functions

Function	Notation	Domain	Range	Equation
Inverse Sine	$y = \arcsin(x)$	$-1 \leq x \leq 1$	$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$	$\sin(y) = x$
Inverse Cosine	$y = \arccos(x)$	$-1 \leq x \leq 1$	$[0, \pi]$	$\cos(y) = x$
Inverse Tangent	$y = \arctan(x)$	$-\infty < x < \infty$	$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$	$\tan(y) = x$