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 inverse functions are essential for solving trigonometric equations and modeling real-world scenarios involving angles.

• Key Point: The six basic trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are not one-to-one, so their inverses are defined on restricted domains.

• Key Point: Inverse trigonometric functions are denoted as arcsin, arccos, arctan, etc.

• Application: Inverse trig functions are used to solve equations and model angles in geometry and physics.

Inverse Sine Function (Arcsine)

Definition and Properties

The inverse sine function, denoted as $y = \arcsin(x)$ or $y = \sin^{-1}(x)$, 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: $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

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

Examples: Evaluating Inverse Sine Without a Calculator

• Example: $\arcsin\left(\frac{\sqrt{3}}{2}\right)$ Find the angle $y$ in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ such that $\sin(y) = \frac{\sqrt{3}}{2}$. Solution: $y = \frac{\pi}{3}$

Examples: Evaluating Inverse Sine With a Calculator

• Example: $\arcsin(-0.46)$ Solution: Use a calculator to find $\arcsin(-0.46) \approx -0.478$ radians.

Inverse Cosine Function (Arccosine)

Definition and Properties

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

• Domain: $-1 \leq x \leq 1$

• Range: $0 \leq y \leq \pi$

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

Inverse Tangent Function (Arctangent)

Definition and Properties

The inverse tangent function, denoted as $y = \arctan(x)$ or $y = \tan^{-1}(x)$, 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: $-\frac{\pi}{2} < y < \frac{\pi}{2}$

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

End Behavior of the Inverse Tangent Function

As $x$ approaches $\infty$, $\arctan(x)$ approaches $\frac{\pi}{2}$. As $x$ approaches $-\infty$, $\arctan(x)$ approaches $-\frac{\pi}{2}$. These are horizontal asymptotes for the function.

Composing Trigonometric and Inverse Trigonometric Functions

Key Properties and Simplification

Composing a trigonometric function with its inverse (or vice versa) can simplify to an algebraic expression, but only when the input is within the restricted domain of the inverse function.

• Always True: $\sin(\arcsin(x)) = x$ for $-1 \leq x \leq 1$

• Always True: $\arcsin(\sin(x)) = x$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

• Similar properties hold for cosine and tangent.

Example: Composing Trig Functions with Arccosine

To simplify $\sin(\arccos(x))$, draw a right triangle with angle $\theta = \arccos(x)$, so $\cos(\theta) = x$. The adjacent side is $x$, the hypotenuse is $1$, and the opposite side is$\sqrt{1 - x^2}$. Thus,$\sin(\arccos(x)) = \sqrt{1 - x^2}$.

Applications of Inverse Trigonometric Functions

Example: Calculating a Viewing Angle

Suppose a photographer is 500 feet from a rising balloon. The angle $\theta$ of elevation to the balloon as a function of its height $s$ is given by $\theta = \arctan\left(\frac{s}{500}\right)$. The rate of change of $\theta$ with respect to $s$ is greater when $s$ is small, and decreases as $s$ increases, due to the horizontal asymptote of the arctangent function.

• Equation: $\theta = \arctan\left(\frac{s}{500}\right)$

• Observation: The angle increases rapidly for small $s$, but more slowly for large $s$.