Trigonometry

“Measurement of triangles”

• Initial side

Angle• Terminal

side

• Vertex Angles are always labeled with either a capital letter or a Greek letter

When the initial side is on the x axis, the angle is in STANDARD POSITION

Types of angles:• Positive angles:

• Negative angles:

• Coterminal angles:

Generated by a counterclockwise rotationGenerated by a clockwise rotation

Angles that have the same initial and terminal sides

There are pictures on page 262

Measures of angles:determined by the amount of rotation

from the initial side to the terminal side• One way to measure are:

RADIANSMeasure of a central angle O that intercepts an

arc s equal in length to the radians r of the

circle

Radians…•Because the circumference of a circle is units it follows that a central angle of one full

revolution (counterclockwise) corresponds to an arc length:

r2

rs 2•A radian measure of a central angle is

r

s

Because the radian measure of an angle of one full revolution is … 2

You need to memorize!!

Q2 Q1

Q3Q4

6

6

13What can you tell me about and ?

Coterminal Angles

2 2To find coterminal

angles:

Practice finding coterminal angles:

3

24

3

Complementary and Supplementary Angles

Complementary: sum = Supplementary: sum = 2

Lets find the complement and supplement of

5

2

5

4Lets find the complement and

supplement of

Day 1 HW

pg. 269 # 1 – 23 odd

Warm Up: Complete the following chart with the correct radian measure Complete the following problems: pg. 269 #’s 4, 6, 8,

12, 14, 18

Degree MeasureThe second way to measure

angles.

What is the relationship between degrees and radians?

Fill in the following circle with the correct radian and degree

measures:

If NO unit measures are specified, then radian

measure is IMPLIED, so assume radians!!Let’s practice converting:

radians2.)560

.)4

180.)3

540.)2

135.)1

Another way to express degrees is to denote fractional parts of

degrees by minutes (‘) and seconds(”).

??? How would represent 64 degrees, 32 minutes, and 47

seconds in this form?

Convert this angle to decimal degree form:

44 15 20

44.255

To convert from decimal degree form to

form, you can use your calculator.

1. Type in decimal degree

2. 2nd angle (apps)

3. Choose 4 (DMS)

4. Enter

Try:

1.) 125.36o 2.) -4.68o

Day 2 HWpg. 269 #’s 24 – 77 by

3’s

APPLICATIONSFinding arc length:

Recall that , therefore s = ?

r

s

*always in radians

Let’s do example 5 on page 267.

s = 2πr

Will be helpful to think of θ: 2π times the number of revolutions

Some more practice:1. Find the angle measure of an angle with a

radius of 10 and arc length of 12.

2. On a circle with a radius of 9 inches, find the length of the arc intercepted by a central angle of 140 degrees.

5

6

inches227

Arc length is used to analyze motion of a particle moving at a constant speed along a circular

path.Linear speed :

Measures how fast the particle moves

Angular speed:

Measures how fast the angle changes

Practice finding linear speed…

The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of this second hand as it passes around the clock face.

Some practice…1. A lawn roller with a 20 inch radius makes 2

revolutions per second. Find the linear speed in inches per second of a point on the outside of the roller after 1 second.

2. A bicycle wheel is turning at a rate of 15 revolutions per minute. The wheel is 24 inches in diameter. Find the linear velocity in feet per second of a piece of gum stuck to the outside of the wheel.

251.32 in/sec

1.57 ft/sec

Practice finding angular speed and linear

speed:A lawn roller with a 10 inch radius makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second, then find the speed of the tractor that is pulling the roller (linear speed).

Some practice:1. The wheel of a truck is turning at a rate of 6

revolutions per second. The wheel is 4 feet in diameter.

a.) Find the angular speed in radians per second.

b.) Find the linear speed, in feet per second, of a point on the rim of the wheel.

12π rad/sec

75.4 ft/sec

HW – Day 3pg. 269 # 25, 29, 35, 37, 41, 49, 51, 53, 55, 57, 63, 67, 79, 81, 83,

87, 99

Day 4 Warm Uppg. 271 # 99

Day 4 – With a partnerpg. 271 # 91, 95, 96,

97, 98, 100, 102

Study problems for quiz…

Pg. 269 # 8, 14, 18, 22, 30, 38, 42, 58, 64,

68, 80, 84, 88