Use your notes from last week:

• Find the value of x and y.

1

45°x

y

A)

Use your notes from last week:

• Find the value of x and y.

1

30°x

y

B)

Use your notes from last week:

• Find the value of x and y.

1

60°x

y

c)

Section 7-1 Measurement of Angles

Objective: To find the measure of an angle in either degrees or radians and

to find coterminal angles.

What we are going to learn in Sec 7.1

• Vocabulary• Angle measure in degrees and radians• Standard position• The critical values on the Unit Circle• Coterminal angles

Section 7-1 Measurement of Angles

• Objective:

1. To find the measure of an angle in either degrees or radians.

2. To find coterminal angles.

Common Terms

• Initial ray is the ray that an angle starts from.

• Terminal ray is the ray that an angle ends on.

• Vertex

Common Terms

• A revolution is one complete circular motion.

Angles in standard position

x

y

Standard Position

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–3

Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248

Vertex at origin

The initial side of an angle in standard position is always located on the positive x-axis.

Angles in standard position

• The vertex of the angle is on (0,0).• Initial ray starts on the positive x-axis• The angle is measure counter clockwise.• The terminal ray can be in any of the

quadrants.

Angle describes the amount and direction of rotation

120° –210°

Positive Angle: rotates counter-clockwise (CCW)

Negative Angle: rotates clockwise (CW)

13

Positive and negative angles

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4

Section 4.1, Figure 4.3, Positive and Negative Angles, pg. 248

When sketching angles, always use an arrow to show direction.

Units of angle measurement

• There are two ways to measure an angle:

Degrees

&

Radians

Units of angle measurement

• There are two ways to measure an angle:

Degree: 1/360th of a circle. That is the measure one sees on a protractor and most people are familiar with.

Angles can be further split into 60 minutes per degree and 60 seconds per minute.

Quadrantal Angle

• If the terminal ray of an angle in standard position lies along an axis the angle is called a quadrantal angle.

• The measure of a quadrantal angle is always a multiple of 90°, or

Quadrantal angles

Standard Position

• When an angle is shown in a coordinate plane, it usually appears in standard position, with its vertex at the origin and its initial ray along the positive x-axis.

Degrees

On one of the circles provided measure 1°

Radian Measure

• Use the string provided to measure the radius.

• Start on the “x-axis” and use the string to measure an arc the same length on the circle.

• The angle created is one radian.

Angle θ is one radian

Units of angle measurement

Radian: when the arc of circle has the same length as the radius of the circle. Angle a measures 1 radian.

1radian

Arc Lenght=radius

Radius

approximations

• 1 radian ~ 57.2958 degrees• 1 degree ~ 0.0174533 radians*• *note: the radian measure is usually stated

as a fraction of .

Sec 7.1 day 2Warm up

• While I check your UC, work on the following:

a) Display the measure of one radian on circle. Display the measure of two radians on a cirlce.

b) Describe what one radian is in terms of the radius of a circle r.

c) Draw a circle and identify a central angle. Describe relationship between central angle and the intercepting arc.

Find the measure of the central angle

The central angle shown has a measure of radian. What is the length of arc ?

Find the measure of the central angle COH

Length CGH = 4 cm

Measure of central angle:

• For radian measure:

s

rFor degree measure:

180 s

rs= arc length

r= radius

30

Radian Measure

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–8

Section 4.1, Figure 4.7, Common Radian Angles, pg. 249

Working with Radians

1180

1𝑟𝑎𝑑𝑖𝑎𝑛=

180 °𝜋

Conversion   formulas :

The conversion process

• When converting between the two units of angle measurement, start with the following template:

Substitute the given measure.

Then solve for missing term.

Example 1Convert 196° to radians

𝜋180 °

=𝑟𝑎𝑑𝑖𝑎𝑛𝐷𝑒𝑔𝑟𝑒𝑒

𝜋180 °

=𝑟𝑎𝑑𝑖𝑎𝑛

196 °

Radian measure is= =

When using th

e radian

measure, expres

s answ

er

as fracti

on of

Convert to degrees.

𝜋180 °

=𝑟𝑎𝑑𝑖𝑎𝑛𝐷𝑒𝑔𝑟𝑒𝑒

𝜋180 °

=

23𝜋

𝑥

𝑥=180 °×

23𝜋

𝜋¿120 °

Mak

e sure

to u

se th

e

degre

e sym

bol.

Calculator

• 2nd APP (Angle)• Use DMS to convert to Degree, minute and

second.• Use Angle to change 40° 20’ to a decimal

value.• For more information click here

Coterminal angles

• Two angles in standard position are called coterminal angles if they have the same terminal ray.

• For any given angle there infinitely many coterminal angles.

Example

• Find two angles, one positive and one negative, that are coterminal with the angle 52°. Sketch all three angles

Solution

Example

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

4

•Sketch all three angles.

x

y

Coterminal Angles generalized:

• Degree measure: θ 360°n • Radian measure: θ 2π n• Where n is a counting number.

Helpful websites

• Trig flash cards• http://mathmistakes.info/facts/TrigFacts/

• Hot math flash cards: • http://hotmath.com/learning_activities/inte

ractivities/trig_flashcard.swf

Homework

• Sec 7.1 Written Exercises• Problems # 1-8 all and • # 9-29 odds• UC with coordinates filled out.

1 degree = 60 minutes

1° = 60

1 minute = 60 seconds

1 = 60

So … 1 degree = _________seconds

3600

Express 365010as decimal degrees

3660

50

3600

10

8361.3636 + .8333 + .00277

OR Use your calculator!! Express 365010as decimal degrees

Enter 36

Press this button ’ ’’Press enter

Enter 50

Press this button ’ ’’Go over to the ’ symbol -- enter

Enter 10Press this button ’ ’’Go over to the ’’ symbol -- enterPress enter

Convert 50 47’ 50’’ to decimal degree

50.7972

Convert 125 27’ 6’’ to decimal degree

125.4517

Can you go backwards and convert the decimal degree to degrees minutes seconds?

Enter 125.4517 Go to DMS hit enter.

Express 50.525 in degrees, minutes, seconds

50º + .525(60) 50º + 36.5

50º + 36 + .5(60)

50 degrees, 36 minutes, 30 seconds