Comprehensive Guide to Advanced Trigonometry

Cosecant

$ext{csc}(x) = \frac{1}{\sin(x)}$, undefined when $\sin(x) = 0$.

Secant

$ext{sec}(x) = \frac{1}{\cos(x)}$, undefined when $\cos(x) = 0$.

Cotangent

$\cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}$, undefined when $\sin(x) = 0$.

Vertical Asymptotes

Occur in reciprocal functions, where the denominator is zero.

Principal Values

Restricted ranges for inverse functions to ensure they are valid functions.

Inverse Sine

$y = \arcsin x$, with input domain $[-1, 1]$ and output range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Inverse Cosine

$y = \arccos x$, with input domain $[-1, 1]$ and output range $[0, \pi]$.

Inverse Tangent

$y = \arctan x$, with input domain $(-\infty, \infty)$ and output range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Pythagorean Identity

$\sin^2 \theta + \cos^2 \theta = 1$.

Tangent/Secant Identity

$1 + \tan^2 \theta = \sec^2 \theta$.

Cotangent/Cosecant Identity

$1 + \cot^2 \theta = \csc^2 \theta$.

Even Functions

Functions that are symmetric about the y-axis, e.g., $\cos(-x) = \cos(x)$.

Odd Functions

Functions that are symmetric about the origin, e.g., $\sin(-x) = -\sin(x)$.

Co-Function Identity

$\sin(x) = \cos\left(x - \frac{\pi}{2}\right)$.

Horizontal Line Test

A method to determine if a function has an inverse; fails for periodic functions.

Solving Trig Equations

Involves isolating a trig function and using identities to solve.

Reference Angle

The acute angle between the terminal side of an angle and the x-axis.

Interval Solutions

Specific solutions provided within a defined range, e.g., $[0, 2\pi]$.

General Solutions

Include all solutions of a trig equation, e.g., $+2\pi n$ or $+\pi n$.

Graphical Behavior of Reciprocal Functions

Where $\sin(x)$ has intercepts, $\csc(x)$ has vertical asymptotes.

Quadrant II Restrictions

For $\arccos$, the resultant angle must be in Quadrant II if the input is negative.

Common Mistake: Inverse Notation

$\sin^{-1}(x)$ means arcsine, not $\frac{1}{\sin(x)}$.

Avoiding Division by a Function

Never divide by a trig function; it can lead to lost solutions.

Periodicity of Functions

Sine and Cosine repeat every $2\pi$; Tangent and Cotangent repeat every $\pi$.

Asymptotes in Graphs

Vertical lines where the function's value approaches infinity.

Hill and Valley Rule

Describes the relationship of peaks in sine/waves to valleys in cosecant.

Domain of Cosecant

$x \neq n\pi$ where $n$ is any integer.

Domain of Secant

$x \neq \frac{\pi}{2} + n\pi$ where $n$ is any integer.