Angles and their Measures

Lesson 1

As derived from the Greek Language, the word trigonometry means “measurement of triangles.”

Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying.

With the development of Calculus and the physical sciences in the 17th Century, a different perspective arose – one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domain.

Consequently the applications expanded to include physical phenomena involving rotations and vibrations, including sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.

We will explore both perspectives beginning with angles and their measures…..

An angle is determined by rotating a ray about its endpoint.

The starting position of called the initial side. The ending position is called the terminal side.

Standard PositionVertex is at the origin, and the initial side is on the x-axis.

Initial Side

Termin

al Sid

e

360,0

90

180

270

III

III IV

Positive Angles are generated by counterclockwise rotation.

Negative Angles are generated by clockwise rotation.

Let’s take a look at how negative angles are labeled on the coordinate graph.

Negative AnglesGo in a Clockwise rotation

45

360,0

90

180

270

Coterminal Angles

Angles that have the same initial and terminal side. See the examples below.

Coterminal AnglesThey have the same initial and terminal sides.

Determine 2 coterminal angles, one positive and one negative for a 60 degree angle.

60

60 + 360 = 420 degrees

60 – 360 = -300 degrees

Give 2 coterminal angles.

30

30 + 360 = 390 degrees

30 – 360 = -330 degrees

Give a coterminal angle, one positive and one negative.

230

230 + 360 = 590 degrees

230 – 360 = -130 degrees

Give a coterminal angle, one positive and one negative.

20

-20 + 360 = 340 degrees

-20 – 360 = -380 degrees

Give a coterminal angle, one positive and one negative.

460 + 360 = 820 degrees

460 – 360 = 100 degrees100 – 360 = -260 degrees

460

Good but not bestanswer.

Complementary AnglesSum of the angles is 90

40 120

Find the complement of each angles:

40 + x = 90

x = 50 degrees

No Complement!

Supplementary AnglesSum of the angles is 180

40 120

Find the supplement of each angles:

40 + x = 180

x = 140 degrees

120 + x = 180

x = 60 degrees

Coterminal Angles:

To find a Complementary Angle:

To find a Supplementary Angle:

360Angle

90 Angle

180 Angle

Radian Measure

One radian is the measure of the central angle, , that intercepts an arc, s, that is equal in length to the radius r of the circle.

So…1 revolution is equal to 2π radians

2C r

2

2

2

2C

r

rC

902

180

3602

radians

radians

radians

Let’s take alook at themplaced on theunit circle.

Radians

Now, let’s add more…..

Radians

22

1

radians4

radians4

3

24

radians4

5radians

4

7

rad28.6 rad14.3

rad57.1

rad71.4

More Common Angles

Let’s take a look at more common angles that are found in the unit circle.

Radians

radians4

radians4

3

radians4

5 radians4

7

rad28.6 rad14.3

Radians

radians4

radians4

3

radians4

5 radians4

7

rad28.6 rad14.3

Look at the Quadrants

Determine the Quadrant of the terminal side of each given angle.

37

122

3371

Q1

Go a little more than one quadrant – negative. Q3

A little more than one revolution. Q1

Determine the Quadrant of the terminal side of each given angle.

14

5156

9

8240

1000

Q3

Q2

2 Rev + 280 degrees. Q4

Coterminal Angles using Radians

Find a coterminal angle.

There are an infinite number of coterminal angles!

Give a coterminal angle, one positive and one negative.

13

4

Give a coterminal angle, one positive and one negative.

5

2

5

Find the complement of each angles:

2

5

Find the supplement of each angles:

2

3

Find the complement & supplement of each angles, if possible:

None

Coterminal Angles:

To find a Complementary Angle:

To find a Supplementary Angle:

2Angle

2Angle

Angle

RECAP

Conversions

radians180

1

1801 radian

4

3

180

135

180135

x

NOTE: The answer is in radians!

2

3

180

270

180270

x

810180

2

9

x

Convert 2 radians to degrees

59.114360180

2

x

The relationship between arc length, radius, and central angle is

Arc Length

Arc Length = (radius) (angle)

1st Change 240 degrees into radians.

radiansx3

4

180

240

180240

inchesors

s

76.163

16

3

44