Unit 6 – Introduction to Trigonometry Degrees and …...Unit 6 { Introduction to Trigonometry Degrees and Radians (Unit 6.2) William (Bill) Finch Mathematics Department Denton High

Unit 6 – Introduction to Trigonometry

Degrees and Radians (Unit 6.2)

William (Bill) Finch

Mathematics DepartmentDenton High School

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

I Understand an angle as a measure of rotation.

I Understand radian and degree measures.

I Be able to convert between radian and degree measure.

I Be able to calculate arc length and sector area.

I Be able to find angular and linear speeds.

W. Finch DHS Math Dept

Radian/Degree 2 / 35


Lesson Goals










Lesson Goals










Lesson Goals










Lesson Goals










Lesson Goals










Angles in Standard Position

An angle in standard position:

I starts on positive x-axis(initial side)

I rotates counter-clockwise forpositive angles

I rotates clockwise for negativeangles

I often named with Greek lettersI theta . . . θI alpha . . .αI beta . . .β

x

y

Initial

Terminal

Positive

x

y

Initial

Terminal

Negative




Degree Measure

x (0◦)

y

0◦

30◦

60◦90◦

120◦

150◦

180◦

210◦

240◦

270◦300◦

330◦

360◦

45◦135◦

225◦ 315◦




Degree-Minutes-Seconds (DMS)

A fraction of a degree can be expressed as a decimal fraction,but historically the degree was divided into minutes (′) andseconds (′′).

1◦ = 60′ and 1′ = 60′′

For example, 32.125◦ = 32◦ 7′ 30′′

Read “ 32 degrees, 7 minutes, and 30 seconds.”




Degree-Minutes-Seconds (DMS)

A fraction of a degree can be expressed as a decimal fraction,but historically the degree was divided into minutes (′) andseconds (′′).

1◦ = 60′ and 1′ = 60′′

For example, 32.125◦ = 32◦ 7′ 30′′

Read “ 32 degrees, 7 minutes, and 30 seconds.”




Example 1

Convert to decimal degrees.

a) 25◦ 15′

b) 12◦ 10′ 33′′




Calculator Instructions – TI-84




Example 2

Convert to degree-minutes-seconds.

a) 48.4◦

b) 21.456◦




Calculator Instructions – TI-84




Radian Measure

One radian is the measure of acentral angle θ that intercepts anarc s equal in length to the radius rof the circle:

θ =s

r

where θ is measured in radians.

x

y

r

srθ

Note that in the diagram above the radius r of the circle is thesame length as the arc s intercepted by the two radii, soθ = 1 rad when s = r .




Radian Measure

The circumference of a circle isone revolution around the circle.

C = 2πr

s = 2πr

s

r= 2π

θ = 2π

θ ≈ 6.28

x

y

θ

A central angle θ that is one revolution is 2π radians.




Radian Measure


C = 2πr

s = 2πr

s

r= 2π

θ = 2π

θ ≈ 6.28

x

y

θ





Radian Measure


C = 2πr

s = 2πr

s

r= 2π

θ = 2π

θ ≈ 6.28

x

y

θ





Radian Measure


C = 2πr

s = 2πr

s

r= 2π

θ = 2π

θ ≈ 6.28

x

y

θ





Radian Measure


C = 2πr

s = 2πr

s

r= 2π

θ = 2π

θ ≈ 6.28

x

y

θ





Radian Measure


C = 2πr

s = 2πr

s

r= 2π

θ = 2π

θ ≈ 6.28

x

y

θ





Radian Measure

One revolution around a circle is slightly more than 6 radians.

x

y

r1 rad2 rad

3 rad

4 rad 5 rad

6 rad

s = r




Radian Measure

x

y

0◦

30◦

60◦90◦

120◦

150◦

180◦

210◦

240◦

270◦300◦

330◦

360◦

45◦135◦

225◦ 315◦

π6

π4

π3

π22π

33π4

5π6

π

7π6

5π4 4π

3 3π2

5π3

7π4

11π6

2π




Special Angles – Learn Them!

x

y

π

x

y

2π

x

yπ2

x

y

3π2





x

yπ4

x

y

5π4

x

y3π4

x

y

7π4





x

yπ3

x

y

4π3

x

y2π3

x

y

5π3





x

y

π6

x

y

7π6

x

y

5π6

x

y

11π6




Radian Measure

θ = π θ = 0

θ =3π

2

θ =π

2 Quadrant I

0 < θ <π

2

(acute angles)

Quadrant IIπ

2< θ < π

(obtuse angles)

Quadrant III

π < θ <3π

2

Quadrant IV3π

2< θ < 2π




Radian-Degree Conversion

Set up and solve this proportion:

radian

degree=π rad

180◦

Hint – always set up the proportion with the unknown anglemeasure in the numerator.




Example 3

Convert to radian measure.

a) 120◦

b) −30◦




Example 4

Convert to degree measure.

a) −3π

4

b)3π

2




Coterminal Angles

Coterminal angles have the same initial and terminal sides.

x

y

α

βx

y

α

β

To find a coterminal angle to some angle θ either add orsubtract a multiple of 2π (or 360◦):

θ ± n · 2π θ ± n · 360◦




Coterminal Angles

Coterminal angles have the same initial and terminal sides.

x

y

α

βx

y

α

β

To find a coterminal angle to some angle θ either add orsubtract a multiple of 2π (or 360◦):

θ ± n · 2π θ ± n · 360◦




Example 5

Sketch the angle given (in radians): θ =2π

3

Then find two coterminal angles: one positive and onenegative.




Example 6

Sketch the angle given (in radians): α = −π4





Example 7

Sketch the angle given (in degrees): β = 25◦





Example 8

Sketch the angle given (in degrees): θ = −150◦





Arc Length

The relationship between a central angle and the length ofthe intercepted arc is

s = rθ

where θ is in radians.

r

s

θ




Example 9

A circle has a radius of 5 inches. Find the length of the arcintercepted by a central angle of 120◦.




Example 10

Winnipeg, Manitoba (Canada) is approximately due north ofDallas. Winnipeg is at a latitude of 49◦ 53′ 0′′N, and Dallas isat a latitude of 32◦ 47′ 39′′N.

Use the given information to find the distance betweenWinnipeg and Dallas (assume the Earth is a perfect spherewith a radius of 4000 miles).




Area of a Sector

A sector of a circle is the region bounded by two radii andtheir intercepted arc.

r

θ

The area of a sector is A =1

2r 2θ (where θ is in radians).




Example 11

A sector has a radius of 12 inches and a central angle of 100◦.Find the area of the sector.




Example 12

Find the approximate area swept bythe wiper blade shown, if the totallength of the windshield wipermechanism is 26 inches.




Linear and Angular Speed

An object moving along an arc has alinear speed given by

ν =arc length

time=

s

t

An object moving along an arc hasan angular speed given by

ω =central angle

time=θ

t

θr

s




Linear and Angular Speed

An object moving along an arc has alinear speed given by

ν =arc length

time=

s

t

An object moving along an arc hasan angular speed given by

ω =central angle

time=θ

t

θr

s




Example 13

A bicycle wheel has a radius of 35 cm.A chalk mark is made on the tire andthen the tire is spun completing one fullrevolution in 0.8 seconds.

a) Determine the linear speed of thechalk mark.

b) Determine the angular speed.




What You Learned

You can now:






I Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,39, 41, 43, 45, 51, 55, 57, 59




What You Learned

You can now:










What You Learned

You can now:










What You Learned

You can now:










What You Learned

You can now:










What You Learned

You can now:










What You Learned

You can now:









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Unit 6 – Introduction to Trigonometry Degrees and …...Unit 6 { Introduction to Trigonometry Degrees and Radians (Unit 6.2) William (Bill) Finch Mathematics Department Denton High