Chapter 4

Trigonometric Functions

AnglesTrigonometry means measurement of triangles.

In Trigonometry, an angle often represents a rotation about a point. Thus, the angle θ shown is the result of rotating its initial ray to its terminal ray.

Standard Position of an AngleAn angle whose vertex is the origin and whose initial side coincides with the positive x-axis is an angles in standard position.

Positive and Negative AnglesPositive angles are generated by counterclockwise rotation.

Negative angles are generated by clockwise rotation.

Labeling angles

Angles are labeled with Greek letters such as alpha ( ) and beta ( ), and theta ( ), as well as uppercase letters.

Quadrantal Angles

If the terminal ray of an angle in standard position lies in the first quadrant, the angle is said to be a first-quadrantal angle. The second-, third- and fourth-quadrant angles are similarly defined.

What is a radian?A common unit for measuring smaller

angles is the degree, of which there are 360° in one revolution.

Illustration of Arc LengthWhen an arc of a circle has the same length as the radius of the circle, the measure of the central angle, is by definition one radian.

One radian is approximately

57.2958 degrees.

Section 4.1, Figure 4.6,Illustration of Six Radian

LengthsThere are 360° or 2 radians in one revolution.

360° = 2 radians180° = radians90° = /2 radians45° = /4 radians

Common Radian Angles

Conversion Formulas1. To convert degrees to radians,

multiply degrees by

2. To convert radians to degrees, mutliply radians by

180

radians

radians180

“Leave in terms of Pi”

Convert 250° to radians

and radians to degrees.4

List the Special Angles

Coterminal AnglesTwo angles are coterminal if they have the same initial and terminal rays.

“Different nameFor the same thing”

Finding Coterminal AnglesYou can find an angle coterminal to a given angle by adding or subtracting ????

For the positive angle find a

Positive coterminal angle.

Negative coterminal angle.

6

13

Finding Coterminal Angles

For the negative angle , find positive and negative coterminal angles.

Find two angles, one positive and one negative, that are coterminal with the angle .

3

2

4

Complementary and Supplementary Angles

Two positive angles are complementary (complements of each other) if their sum is 90° or

radians.

Two positive angles are supplementary (supplements of each other) if their sum is 180° or

radians.

2

Arc length

Arc length is measured in radians by the formula:

srWhere S is arc length

r is the radius is the central angle

A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°.

A circle has a radius of 9 feet. Find the length of the arc intercepted by a central angle of /4.

Linear and Angular SpeedImagine a particle moving at a constant speed along

the edge of a circle. The distance the particle travels with respect to time is the particle’s linear speed.

Linear speed =

Linear speed measures how fast the particle moves.

ArcLength

Times

t

Linear and Angular SpeedImagine a spinning circle. Consider a central angle of

that circle.

Angular speed measures how fast the angle changes per a unit of time.

Angular speed =

Angular speed measures how fast the angle changes.

CentralAngle

Time

t

See examples on page 254 for Linear and Angular speed.

Goes back to this idea

Classwork…

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