Trigonometric Functions and Their Properties

Function Structure: Functions of the form ( f(x) = A \cos(Bx - C) + D ) and ( f(x) = A \sin(Bx - C) + D ) represent sinusoidal graphs.
Example Functions:
- ( y = \cos(3x) )
- Properties:
- Amplitude: 1 (the coefficient of cosine)
- Period: ( 2\pi/3 ) (from the factor 3 in front of ( x ))
- Phase Shift: 0 (since there is no horizontal shift)
- ( y = -6 \cos{(\frac{\pi}{4}(x + \frac{1}{2}))} )
- Amplitude: 6
- Period: ( \frac{2\pi}{\frac{\pi}{4}} = 8 )
- Phase Shift: ( -\frac{1}{2} )

Amplitude is the height from the middle value (equilibrium) to the peak (or trough) of a wave.
Period is the length of one complete cycle of the function, calculated as ( \frac{2\pi}{B} ).
Phase Shift is found using ( \frac{C}{B} ).

Form of function for a given graph is described as ( f(x) = A \cos(Bx - C) + D ).
Example transformation:
- Write a function given an amplitude of 4, phase shift of 3π, and period of π.
Example output: ( y = 4 \sin(6x - 3\pi) + 5 ) or ( y = -4 \sin(6x - 3\pi) + 5 )

Function Examples:
- ( f(x) = \sec(x) ) and ( f(x) = \tan(x) ) will have distinct characteristics when graphed with vertical asymptotes at points where the function is undefined.
Behavior Near Asymptotes:
- As x approaches ( \frac{-\pi}{2} ), ( f(x) \to \infty ).
- Use of limits helps describe behavior approaching these critical points.

Function Definitions:
- ( \cos^{-1}(x) ) is the inverse cosine function or arccos.
- ( \tan^{-1} ) represents the inverse tangent.
Calculating Results:
- For example, find ( \sin^{-1}(\frac{1}{2}) ) which equals ( \frac{\pi}{6} ).
Approximations: Use calculators for values requiring decimal values for angles.
Domain of Inverse Functions: Varies depending on whether the function is including all real numbers or if it's constrained to specific intervals like ([0, \pi]\, [ -\frac{\pi}{2}, \frac{\pi}{2}]).