Precalculus Study Guide: Trigonometric Functions and Their Applications (Sections 5.4–5.8)

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Trigonometric Functions of Real Numbers

Unit Circle and Trigonometric Definitions

The unit circle is a fundamental concept in trigonometry, defined as a circle with radius 1 centered at the origin. It allows us to define trigonometric functions for all real numbers, not just angle measures. If t is a real number and P = (x, y) is the point on the unit circle corresponding to t, the trigonometric functions are defined as follows:

sin t = y
cos t = x
tan t = \frac{y}{x},\ x \neq 0
csc t = \frac{1}{y},\ y \neq 0
sec t = \frac{1}{x},\ x \neq 0
cot t = \frac{x}{y},\ y \neq 0

Unit circle trigonometric definitions

Domain and Range:

Function	Domain	Range
sin x	All real numbers	[−1, 1]
cos x	All real numbers	[−1, 1]
tan x	All real numbers except odd multiples of	All real numbers
sec x	All real numbers except odd multiples of
csc x	All real numbers except multiples of
cot x	All real numbers except multiples of	All real numbers

Even/Odd and Periodic Properties

Trigonometric functions exhibit symmetry and periodicity:

Even functions: ,
Odd functions: , , ,
Periodicity: , , , , ,

Graphs of Sine and Cosine

Key Properties and Transformations

The graphs of sine and cosine functions are periodic and exhibit specific symmetries:

y = sin x: Domain: , Range: [−1, 1], Period: , Symmetry: odd
y = cos x: Domain: , Range: [−1, 1], Period: , Symmetry: even

Graph of y = sin x Graph of y = cos x

Transformations:

General form: or
Amplitude:
Period:
Phase shift:
Key points transformation: , ,

Graphs of Tangent and Cotangent

Key Properties and Graphs

The tangent and cotangent functions have distinct domains, ranges, and periods:

y = tan x: Domain: all real numbers except odd multiples of , Range: , Period: , Symmetry: odd
y = cot x: Domain: all real numbers except multiples of , Range: , Period: , Symmetry: odd

Graph of y = tan x Graph of y = cot x

Key points for tangent:

x	y
	Asymptote
	−1
0	0
	1
	Asymptote

Key points for cotangent:

x	y
0	Asymptote
	1
	0
	−1
	Asymptote

Inverse Trigonometric Functions

Restricting Domains and Evaluating Inverses

Inverse trigonometric functions are defined by restricting the domains of the original functions to make them one-to-one. This ensures each input has a unique output.

Function	Restricted Domain (Angles)	Range (Ratios)
y = sin x
y = cos x
y = tan x

Inverse function notation:

(arcsin)
(arccos)
(arctan)

Types of questions:

Evaluate inverse functions for common ratios (e.g., )
Recognize when the inverse cannot be evaluated if the ratio is out of domain

Trig Function Composition

Types of Trigonometric Compositions

Compositions of trigonometric and inverse trigonometric functions require careful attention to domains and ranges:

(only if is in the domain of ; otherwise undefined)
(evaluate using the unit circle)
(assign an angle, draw a triangle using SOHCAHTOA, determine quadrant, and find the correct ratio)
(evaluate using the unit circle)

Applications of Trigonometric Functions

Solve a Right Triangle

Solving right triangles involves finding unknown sides and angles using trigonometric relationships. The triangle has three sides (a, b, c) and three angles (A, B, C), with C typically representing the right angle.

Angle and side relationships: Each angle is opposite its corresponding side.
Given three out of six values: Find the missing three using trigonometric tools.
Tools:
- (Pythagorean Theorem)
- SOHCAHTOA:

Right triangle with sides and angles labeled

Solving application problems:

If a diagram is not given, create one.
Locate the right triangles.
Pick a tool from above that leads to the answer.
Use algebra and trigonometry to solve the problem.