Inverse Trigonometric Functions: Definitions, Properties, and Applications

Inverse Trigonometric Functions

Introduction to Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine angles when given trigonometric ratios. Since the six basic trigonometric functions are periodic and not one-to-one, their domains must be restricted to define inverses. These functions are essential for solving trigonometric equations and modeling real-world scenarios involving angles.

• Key Point 1: The six basic trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are not one-to-one over their entire domains due to periodicity.

• Key Point 2: By restricting the domain to intervals where the function is one-to-one, we can define inverse functions for sine, cosine, and tangent.

• Key Point 3: Inverse trigonometric functions are denoted as arcsin, arccos, and arctan, or as sin-1, cos-1, and tan-1.

Inverse Sine Function (Arcsine)

Definition and Properties

The inverse sine function, or arcsine, gives the angle whose sine is a given number. To ensure the function is one-to-one, the domain of the sine function is restricted to $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$.

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

• Domain: $x \in [-1, 1]$

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

Evaluating Inverse Sine Without a Calculator

• Key Point: To find $\arcsin(x)$ for special values, locate the corresponding angle on the unit circle whose sine is $x$.

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

Evaluating Inverse Sine With a Calculator

• Key Point: Calculators can be used to approximate values of $\arcsin(x)$ for non-special values.

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

Inverse Cosine Function (Arccosine)

Definition and Properties

The inverse cosine function, or arccosine, gives the angle whose cosine is a given number. The domain of the cosine function is restricted to $[0, \pi]$ to ensure it is one-to-one.

• Definition: $y = \arccos(x)$ means $\cos(y) = x$ and $y \in [0, \pi]$.

• Domain: $x \in [-1, 1]$

• Range: $y \in [0, \pi]$

Inverse Tangent Function (Arctangent)

Definition and Properties

The inverse tangent function, or arctangent, gives the angle whose tangent is a given number. The domain of the tangent function is restricted to $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$ to ensure it is one-to-one.

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

• Domain: $x \in (-\infty, \infty)$

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

End Behavior of the Inverse Tangent Function

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

Composing Trigonometric and Inverse Trigonometric Functions

Fundamental Identities

Compositions of trigonometric and inverse trigonometric functions can often be simplified to algebraic expressions. The following identities are always true when defined:

• $\sin(\arcsin(x)) = x$ for $x \in [-1, 1]$

• $\arcsin(\sin(x)) = x$ for $x \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$

• Similar identities hold for cosine and tangent, with their respective restricted domains and ranges.

Example: Composing Trig Functions with Arccosine

• Key Point: To simplify $\sin(\arccos(x))$, draw a right triangle with angle $\theta = \arccos(x)$, so $\cos(\theta) = x$.

• Example: In a right triangle with adjacent side $x$ and hypotenuse 1$, so$\sin(\arccos(x)) = \sqrt{1 - x^2}$.

Applications of Inverse Trigonometric Functions

Example: Calculating a Viewing Angle

Inverse trigonometric functions are used in real-world applications, such as determining the angle of elevation to an object. For example, a photographer observing a rising balloon can use the arctangent function to find the angle of elevation based on the balloon's height and horizontal distance.

• Key Point: If the balloon is $s$ feet above the ground and the observer is 500 feet away, the angle $\theta$ is given by $\theta = \arctan\left(\frac{s}{500}\right)$.

• Observation: 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.

Summary Table: Inverse Trigonometric Functions

Function	Definition	Domain	Range
$y = \arcsin(x)$	$\sin(y) = x$	$[-1, 1]$	$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$
$y = \arccos(x)$	$\cos(y) = x$	$[-1, 1]$	$[0, \pi]$
$y = \arctan(x)$	$\tan(y) = x$	$(-\infty, \infty)$	$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$