6.1 Angles

Definition and Types of Angles

An angle is formed by two rays (the sides of the angle) sharing a common endpoint (the vertex). Angles are fundamental in trigonometry and are measured in degrees or radians.

• Central Angle: An angle whose vertex is at the center of a circle.

• Angle in Standard Position: An angle with its vertex at the origin and its initial side along the positive x-axis.

Radians and Degrees

The radian is the standard unit of angular measure in mathematics. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

• Conversion between degrees and radians:

Arc Length and Area of a Sector

• Arc Length: The length of an arc of a circle of radius subtended by a central angle (in radians) is:

• Area of a Sector: The area of a sector of a circle of radius and central angle (in radians) is:

Linear and Angular Speed

• Linear Speed (): The rate at which an object moves along a circular path.

• Angular Speed (): The rate at which the central angle changes, measured in radians per unit time.

6.2 Trig Functions: Unit Circle

Coterminal and Reference Angles

• Coterminal Angles: Angles that share the same initial and terminal sides but may have different measures. They differ by integer multiples of or radians.

• Reference Angle: The acute angle formed by the terminal side of a given angle and the x-axis.

Definition of the Six Trigonometric Functions

For a point on the terminal side of an angle in standard position, and :

• (if )

• (if )

• (if )

• (if )

On the unit circle (), these simplify to , , , etc.

Exact Values for Special Angles

• Know the exact values of trigonometric functions for quadrantal angles (, , , ) and for multiples of and .

Example Table:

6.3 Properties of the Trig Functions

Domain, Range, Period, and Parity

• Domain and Range:

• Signs in Quadrants: The sign of each function depends on the quadrant in which the terminal side of the angle lies.

All Students Take Calculus mnemonic: In Quadrant I, all are positive; II: sine positive; III: tangent positive; IV: cosine positive.

• Even/Odd Functions: , , , are odd; , are even.

Determining Quadrants and Values

• Given the sign of a trigonometric function and the value of one function, you can determine the quadrant and compute the exact values of the other functions using identities.

Example: If and is in Quadrant II, then (using ).

6.4–6.6 Graphing Trig Functions

General Form and Key Features

The general form for the sine function is:

• Amplitude (): The maximum value from the midline; .

• Period (): The length of one cycle; .

• Phase Shift (): Horizontal shift; .

• Vertical Shift (): Moves the graph up or down.

Graphing Steps

1. Identify amplitude, period, phase shift, and vertical shift.

2. Plot the midline (at ).

3. Mark key points for at least two periods.

4. Apply transformations in the correct order: horizontal shift, stretch/compression, vertical shift.

Finding Equations from Graphs

• Given a graph, determine amplitude, period, and phase shift to write the equation in the form or .

Example: If a sine wave has amplitude 2, period , phase shift to the right, and midline at , its equation is: