13.2

General Angles and Radian Measure

History Lesson of the Day

Hippocrates of Chois (470-410 BC) and Erathosthenes of Cyrene (276-194 BC) began using triangle ratios that were used by Egyptian and Babylonian engineers 4000 years earlier.

Term “trigonometry” emerged in the 16th century from Greek roots.

“Tri” = three

“gonon” = side

“metros” = measure

Why Study Trig?

• Trig functions arose from the consideration of ratios within right triangles.– Ultimate tool for engineers in the ancient world.

• As knowledge progressed from a flat earth to a world of circles and spheres, trig became the secret to understanding circular phenomena.

• Circular motion led to harmonic motion and waves.– Electrical current– Modern telecommunications– Store sound wave digitally on a CD

What you’ll learn about

• The Problem of Angular Measure• Degrees and Radians• Arc Length• Area of a Sector

… and whyAngles are the domain elements of the

trigonometric functions.

Why 360º?The idea of dividing a circle into 360 equal pieces dates The idea of dividing a circle into 360 equal pieces dates back to the sexagesimal (60-based) counting system of back to the sexagesimal (60-based) counting system of the ancient Sumerians. Early astronomical calculations the ancient Sumerians. Early astronomical calculations linked the sexagesimal system to circles.linked the sexagesimal system to circles.

The problem is that degree The problem is that degree units have no mathematical units have no mathematical relationship whatsoever to relationship whatsoever to linear units. Therefore we linear units. Therefore we needed another way to needed another way to measure a circle.measure a circle.

CircumferenceCircumference: The : The distancedistance all the way around all the way around a circle. a circle.

Vocabulary

rC 2What is the circumference of a circle whose What is the circumference of a circle whose radius equals 10 feet?radius equals 10 feet?

inchesinchesC .20)10(**2

Your Turn:rC 21.1. Circle #1 has a radius of 1 inch, what is the Circle #1 has a radius of 1 inch, what is the circumference of circle #1? circumference of circle #1? (leave ‘pi’ in your answer)(leave ‘pi’ in your answer)

2. Circle #2 has a radius of 3 inches, what is the 2. Circle #2 has a radius of 3 inches, what is the circumference of circle #2? circumference of circle #2? (leave ‘pi’ in your answer)(leave ‘pi’ in your answer)

3.3. What does the What does the ratioratio the circumference of the circumference of Circle #1 to Circle #2?Circle #1 to Circle #2?

The The ratioratio of circumference to of circumference to radiusradius is is CONSTANT.CONSTANT.

2r

Cregardless of the size of the circle.regardless of the size of the circle.

Vocabulary

2

Degree measure of a circle: 360Degree measure of a circle: 360ºº

Radian measurement of a circle = Radian measurement of a circle =

Radiansinches

inches

inchesr

inchesC..2..2

..

..

Radian measureRadian measure: the ratio of the arc length to : the ratio of the arc length to the radius of the circle:the radius of the circle:

radius

lengtharcMeasureRadian

....

Degree-Radian Conversion Degree-Radian Conversion

?180

o

180To convert radians to degrees, multiply by .

radians radians

To convert degrees to radians, multiply by .180

180° = π radians

?180

o

These areThese are “ “conversion factorsconversion factors””

When you multiply a number by oneWhen you multiply a number by one of these factors, it converts the units.of these factors, it converts the units.

Converting from Degrees to Radian Measure

140°

2

o180

Converting from Radian Measure to Degrees

o180

What propertyWhat property is used here?is used here?

9

7

180

140

90

Your Turn: Convert between radians and degrees.

4.

5.

?3

11

?240 o

Initial Side, Terminal Side

beginning position of the raybeginning position of the ray

final position of the rayfinal position of the ray

VertexVertex

αα, , ββ, , θθ = the measure of the angle= the measure of the angle

VocabularyStandard Position An acute angle with one ray along the x-axis and the other

ray rotated clockwise from the first ray.

Terminal SideTerminal Side

In Trigonometry, weIn Trigonometry, we sometimes use a circlesometimes use a circle with the vertex of the with the vertex of the angle at the center ofangle at the center of the circle.the circle.

VocabularyStandard Position An acute angle with one ray along the positive x-axis and the other

ray rotated clockwise from the first ray.

Initial SideInitial Side

Terminal SideTerminal Side

In Trigonometry, weIn Trigonometry, we sometimes use a circlesometimes use a circle with the vertex of the with the vertex of the angle at the center ofangle at the center of the circle.the circle.

We use the measure of the acute angle with the x-axis.

VocabularyStandard Position An acute angle with one ray along the positive x-axis and the other

ray rotated clockwise from the first ray.

Initial Side

Terminal Side

We use the measure of the acute angle with the x-axis.

Angle measures

00ºº

9090ºº

180180ºº

270270ºº

Draw the angle with the given Draw the angle with the given measure in standard form.measure in standard form.

220220ºº

220º

220º

4040ºº

4040ºº

Your turn:

Draw an angle in Draw an angle in standard positionstandard position that that has a measure of:has a measure of:

6. 1356. 135ºº

7. 2907. 290ºº

Co-terminal Angles

45º

What is the difference inWhat is the difference in position on the position on the unit circleunit circle if terminal side stops atif terminal side stops at 4545º or goes all the wayº or goes all the way around and stops at 405º ?around and stops at 405º ?

Co-terminal AnglesWhat is the difference inWhat is the difference in position on the position on the unit circleunit circle if terminal side stops atif terminal side stops at 4545º or goes all the wayº or goes all the way around and stops at 405º ?around and stops at 405º ?

45º

There is no difference !!There is no difference !!

Although the angular Although the angular measure is different theymeasure is different they are are co-terminal angles.co-terminal angles.

Finding Co-terminal Angles

Find a Find a positivepositive and a and a negativenegative angle that are angle that are co-terminalco-terminal with 45 with 45°.°.

We’ve already found We’ve already found one positive co-terminalone positive co-terminal angle with 45angle with 45° (405°).° (405°).

Can you find another?Can you find another?

405405° + 360° = 765° ° + 360° = 765°

Negative angleNegative angle::4545°- 360° = -315° °- 360° = -315°

45º

Your Turn:

8.8.

9.9.

Find a positive co-terminal angle with 120Find a positive co-terminal angle with 120°°

Find a negative co-terminal angle with 270Find a negative co-terminal angle with 270°°

Finding Co-terminal Angles

6

Find a positive and negative co-terminal angle Find a positive and negative co-terminal angle with: with:

Notice the angle measure is now in Notice the angle measure is now in radiansradians..

6

132

6

6

112

6

Your Turn:

10.10.

Find a negative co-terminal angle withFind a negative co-terminal angle with

3

2

11.11.

Find a positive co-terminal angle withFind a positive co-terminal angle with

2

Arc Length

rs r

RememberRemember:: radian measure is theradian measure is the ratio of arc length to radius.ratio of arc length to radius.

Which gives us thisWhich gives us this formula for arc length.formula for arc length.

GothchaGothcha: the angle: the angle measure measure mustmust be be in radians not degrees.in radians not degrees.

Arc LengthRadian = Radian =

'

..

radius

lengtharc

= Greek letter “theta”= Greek letter “theta” OftenOften used as a used as a variablevariable to to denote the measure ofdenote the measure of

an angle.an angle.rs ““arc length = radius * angle measure (in radians)”arc length = radius * angle measure (in radians)”

r = 5 inchesr = 5 inches radians3

Arc length = ?Arc length = ?

radiansinchess

3)5(

inchess

3

5

Your Turn:

12. radius = 10 inches, 12. radius = 10 inches, 7

3 Arc length = ?Arc length = ?

13. Arc length = 13. Arc length = 3

2Radius = 6 inchesRadius = 6 inches

What is the angle measure (in radians)?What is the angle measure (in radians)?

14. What is the angle measure for problem14. What is the angle measure for problem #12 in degrees? #12 in degrees?

Sector Area

10 ft

Area of a circleArea of a circle::

What fraction of the circleWhat fraction of the circle is a 30is a 30º sector?º sector?

rA 2

3030ºº

ftA .20

121

360

30..

circleofFraction

Sector Area = Sector Area = ).20(121 ft

Sector Area = Sector Area = ft3

5

Your turn:

14 inch pizza (diameter)14 inch pizza (diameter) Slice is 1/8 of the pizzaSlice is 1/8 of the pizza

15. Find the area of a slice of pizza.15. Find the area of a slice of pizza.

HOMEWORK