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 ratio. Since the six basic trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are periodic and not one-to-one over their entire domains, we restrict their domains to intervals where they are one-to-one in order to define their inverses.

• Inverse functions are denoted as arcsin, arccos, and arctan for sine, cosine, and tangent, respectively.

• These functions are essential for solving trigonometric equations and for applications 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: $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

• The function is one-to-one on this restricted domain.

Evaluating Inverse Sine Without a Calculator

• To find $\arcsin(x)$ for special values, use the unit circle and reference triangles.

• Example: $\arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$, since $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ and $\frac{\pi}{3}$ is in the range $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$.

Evaluating Inverse Sine With a Calculator

• For non-special values, use a calculator set to radian mode.

• Example: $\arcsin(-0.46) \approx -0.478$ 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: $0 \leq y \leq \pi$

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: $-\frac{\pi}{2} < y < \frac{\pi}{2}$

End Behavior of the Inverse Tangent Function

• 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, a photographer observing a hot-air balloon can use the inverse tangent function to calculate the angle of elevation based on the balloon's height and horizontal distance.

• If the balloon is $s$ feet above the ground and the observer is 500 feet away, $\tan\theta=\frac{s}{500}$meaning that the angle $\theta$ 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 asymptotes of the arctangent function.

Summary Table: Inverse Trigonometric Functions

Function	Notation	Domain	Range
Inverse Sine	$y = \arcsin(x)$	$-1 \leq x \leq 1$	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$
Inverse Cosine	$y = \arccos(x)$	$-1 \leq x \leq 1$	$0 \leq y \leq \pi$
Inverse Tangent	$y = \arctan(x)$	$-\infty < x < \infty$	$-\frac{\pi}{2} < y < \frac{\pi}{2}$