Analytic Trigonometry: Inverse Sine, Cosine, and Tangent Functions

Analytic Trigonometry

Inverse Sine, Cosine, and Tangent Functions

This section introduces the inverse trigonometric functions, their definitions, properties, and applications in solving equations and evaluating composite functions. Understanding these concepts is essential for solving trigonometric equations and analyzing functions in Precalculus.

Inverse Sine Function

Definition and Properties

The inverse sine function, denoted as $y = \sin^{-1} x$ or $y = \arcsin x$, is the function that gives the angle $y$ whose sine is $x$. The domain of the inverse sine function is $[-1, 1]$, and its range is $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$.

• Definition: $y = \sin^{-1} x$ if and only if $\sin y = x$ and $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.

• Domain: $-1 \leq x \leq 1$

• Range: $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

Finding Exact Values of Inverse Sine

• To find $\sin^{-1}(x)$, determine the angle $\theta$ in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ such that $\sin \theta = x$.

• Use the unit circle or reference tables for common values.

$\theta$	$-\frac{\pi}{2}$	$-\frac{\pi}{3}$	$-\frac{\pi}{4}$	$-\frac{\pi}{6}$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin \theta$	$-1$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{2}}{2}$	$-\frac{1}{2}$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$

Finding Approximate Values of Inverse Sine

• For values not found in the table, use a calculator set to radians or degrees as required.

• Round answers as specified (e.g., to two decimal places).

Inverse Cosine Function

Definition and Properties

The inverse cosine function, denoted as $y = \cos^{-1} x$ or $y = \arccos x$, gives the angle $y$ whose cosine is $x$. The domain is $[-1, 1]$, and the range is $[0, \pi]$.

• Definition: $y = \cos^{-1} x$ if and only if $\cos y = x$ and $0 \leq y \leq \pi$.

• Domain: $-1 \leq x \leq 1$

• Range: $0 \leq y \leq \pi$

Finding Exact Values of Inverse Cosine

• To find $\cos^{-1}(x)$, determine the angle $\theta$ in $[0, \pi]$ such that $\cos \theta = x$.

• Use reference tables for common values.

$\theta$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$\pi$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{3}}{2}$	$-1$

Inverse Tangent Function

Definition and Properties

The inverse tangent function, denoted as $y = \tan^{-1} x$ or $y = \arctan x$, gives the angle $y$ whose tangent is $x$. The domain is $(-\infty, \infty)$, and the range is $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$.

• Definition: $y = \tan^{-1} x$ if and only if $\tan y = x$ and $-\frac{\pi}{2} < y < \frac{\pi}{2}$.

• Domain: $-\infty < x < \infty$

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

Finding Exact Values of Inverse Tangent

• To find $\tan^{-1}(x)$, determine the angle $\theta$ in $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$ such that $\tan \theta = x$.

• Use reference tables for common values.

$\theta$	$-\frac{\pi}{2}$	$-\frac{\pi}{3}$	$-\frac{\pi}{4}$	$-\frac{\pi}{6}$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\tan \theta$	Undefined	$-\sqrt{3}$	$-1$	$-\frac{1}{\sqrt{3}}$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	Undefined

Properties of Inverse Trigonometric Functions

Key Properties

• For $x$ in the domain of $\sin^{-1}$, $\sin(\sin^{-1} x) = x$.

• For $y$ in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$, $\sin^{-1}(\sin y) = y$.

• For $x$ in the domain of $\cos^{-1}$, $\cos(\cos^{-1} x) = x$.

• For $y$ in $[0, \pi]$, $\cos^{-1}(\cos y) = y$.

• For $x$ in the domain of $\tan^{-1}$, $\tan(\tan^{-1} x) = x$.

• For $y$ in $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$, $\tan^{-1}(\tan y) = y$.

Examples of Composite Functions

• Evaluate $\sin(\sin^{-1} x)$ for $x$ in $[-1, 1]$; the result is $x$.

• Evaluate $\sin^{-1}(\sin y)$ for $y$ in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$; the result is $y$.

• For values outside the principal interval, adjust using properties of even/odd functions or periodicity.

Finding the Inverse of a Trigonometric Function

Procedure

• Given a function $f(x)$ involving a trigonometric function, interchange $x$ and $y$ and solve for $y$.

• Express the result in terms of the appropriate inverse trigonometric function.

• Determine the domain and range of the original and inverse functions using the definitions above.

Solving Equations Involving Inverse Trigonometric Functions

General Steps

• Isolate the inverse trigonometric function on one side of the equation.

• Apply the definition to rewrite the equation in terms of the corresponding trigonometric function.

• Solve for the variable, considering the domain and range restrictions.

Additional info: The above notes are based on the standard Precalculus curriculum and the provided textbook content. All tables and graphs are reconstructed for clarity and completeness. Examples and properties are expanded for academic rigor and exam preparation.