Trigonometric Functions of Real Numbers

Unit Circle and Trigonometric Definitions

The unit circle is a circle with radius 1 centered at the origin. It is fundamental for defining 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, then:

• 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

Domain and Range of the six trigonometric functions:

Even/Odd and Periodic Properties

Trigonometric functions have symmetry properties:

• Even functions: ,

• Odd functions: , , ,

Periodicity:

• Period : , , ,

• Period : ,

Graphs of Sine and Cosine (Section 5.5)

Key Properties and Transformations

The graphs of y = sin x and y = cos x are periodic and fundamental to trigonometry.

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

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

Transformations: For or :

• Amplitude:

• Period:

• Phase shift (horizontal shift):

To graph a transformed function, apply the following steps to the key points:

• Multiply y-values by A

• Divide x-values by B (reciprocal of B)

• Shift x-values by (phase shift)

Key points for y = sin x:

Key points for y = cos x:

Types of Problems:

• Given an equation, graph using final key points

• Given information about the graph, find the equation

• Given the graph, find the equation

Graphs of Tangent and Cotangent (Section 5.6)

Key Properties and Graphs

The tangent and cotangent functions have unique properties and graphs, including vertical asymptotes.

• 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

Key points for y = tan x:

Key points for y = cot x:

Types of Questions: Be able to graph and using key points and asymptotes.

Inverse Trigonometric Functions (Section 5.7)

Restricting Domains and Evaluating Inverses

To make trigonometric functions one-to-one (so each input has a unique output), their domains are restricted:

For inverse functions:

Types of Questions:

• Evaluate sine and cosine inverses for ratios: (positive and negative)

• Evaluate tangent inverse for ratios: (positive and negative)

• Recognize when the inverse cannot be evaluated if the ratio is out of domain

Trig Composition Types:

1. if is in the domain of ; otherwise, undefined.

2. : Evaluate using the unit circle.

3. : Assign an angle , draw a triangle, determine the quadrant, and use SOHCAHTOA to find the correct ratio.

4. : Evaluate using the unit circle.

Applications of Trigonometric Functions (Section 5.8)

Solving Right Triangles and Application Problems

To solve a right triangle, use the following tools:

• Three sides:

• Three angles: (C is the right angle unless otherwise noted)

• Matching sides and angles are opposite each other

• Given three out of six parts, find the missing three

Key Tools:

2. (Pythagorean Theorem)

3. SOHCAHTOA (definitions of sine, cosine, tangent in right triangles)

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