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.