Rev.S08

Review 5.1-5.3

Radian Measure and

Circular Functions

2

How to Convert Between Degrees and Radians?

1. Multiply a degree measure by radian and

simplify to convert to radians.

2. Multiply a radian measure by and simplify

to convert to degrees.

3

Example of Converting from Degrees to Radians

Convert each degree measure to radians. a)60

b) 221.7

€

739π

600OR

4

Example of Converting from Radians to Degrees

Convert each radian measure to degrees.

a)

b) 3.25

5Rev.S08

Let’s Look at Some Equivalent Angles in Degrees and Radians

6.282π3601.0560

4.71270.7945

3.14π180.5230

1.5790000

ApproximateExactApproximateExact

RadiansDegreesRadiansDegrees

6

Let’s Look at Some Equivalent Angles in Degrees and Radians (cont.)

7

Examples Find each function value. a) b)

€

=− 3

2−1

2= − 3

2 • −21 = 3

€

=− 3

2

IF YOUR ANGLE IS ON THE UNIT CIRLE THEN USE IT!!

€

tanθ =y

x

Or you can set up your triangles!

€

− 3

2€

−1

2

€

−1

2

€

− 3

2

tan = opp adj

sin = opp hyp

θθ

1

1

8

How to Find Arc Length of a Circle?

The length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or s = rθ, θ in radians.

9

Example of Finding Arc Length of a Circle A circle has radius 18.2

cm. Find the length of the arc intercepted by a central angle having each of the following measures.

a)

b) 144

10

Example of Application

A rope is being wound around a drum with radius .8725 ft. How much rope will be wound around the drum it the drum is rotated through an angle of 39.72?

Convert 39.72 to radian measure.

11

Let’s Practice Another Application of Radian Measure Problem

Two gears are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates through 225, through how many degrees will the larger gear rotate?

12

Let’s Practice Another Application of Radian Measure Problem (cont.)

Find the radian measure of the angle and then find the arc length on the smaller gear that determines the motion of the larger gear.

13

Let’s Practice Another Application of Radian Measure Problem (cont.)

An arc with this length on the larger gear corresponds to an angle measure θ, in radians where

Convert back to degrees.

14

How to Find Area of a Sector of a Circle?

A sector of a circle is a portion of the interior of a circle intercepted by a central angle. “A piece of pie.”

The area of a sector of a circle of radius r and central angle θ is given by

15

Example

Find the area of a sector with radius 12.7 cm and angle θ = 74.

Convert 74 to radians.

Use the formula to find the area of the sector of a circle.

16

What is a Unit Circle? A unit circle has its center at the origin and a

radius of 1 unit.

Note: r = 1

s = rθ,s=θ in radians.

17

Circular Functions and their Reciprocals

x

y1

y

yyθ

This is an example of a triangle in the 1st quadrant

Remember our two special triangles that make up the unit cirlce:

19

Let’s Look at the Unit Circle Again

Because its made up of our “special” triangles.

20

Example of Finding Exact Circular Function Values

Find the exact values of Evaluating a circular function at the real number

is equivalent to evaluating it at radians. An angle of intersects the unit circle at the point .

Since sin θ = y, cos θ = x, and

IF AN ANGLE IN STANDARD POSITION MEASURES THE GIVEN RADIANS, DETERMINE WHICH QUADRANT IT’S TERMINAL SIDE LIES.

€

7π

12

−2π

3

371°

14π

5

II

III

I

II

Change the given degree measure to radian measure in terms of π.

€

36°

−250°

−145°

6°

€

•π

180

€

•π

180

€

•π

180

€

•π

180

€

=3π

10

€

=−25π

18

€

=−29π

36

€

=π30

Change the given radian measure into degrees.

€

−1

4π

3π

16

−7π

9

€

•180

π

€

•180

π

€

•180

π

€

•180

π

=-57.3°

=720°

=33.75°

=-140°

Find one positive and one negative angle that is coterminal with an angle measuring the given θ

€

70°

−2π

5

−300°

3π

4

--290°, 430°

--660°, 60°€

8π

5,−12π

5

€

11π

4,−5π

4

add 360°subtract 360°

add 360°subtract 360°

add 2πsubtract 2π

add 2πsubtract 2π

Find the reference angle for the angle given:

€

−20°

160°

10π

3

−5π

8

20°

20°

€

4π

3−

3π

3=π

3

€

π −5π

88π

8−

5π

8=

3π

8

Is the acute angle formed with the x-axis

θ

θ

θ

one full revolutionWith left over

€

4π

3

θ

Find the length of an arc that subtends an angle given, in a circle with diameter 20 cm. Write your answer to the nearest tenth

€

π6

π

3

€

90°

36°€

s = (10)π

6

⎛

⎝ ⎜

⎞

⎠ ⎟=

€

s = (10)π

3

⎛

⎝ ⎜

⎞

⎠ ⎟=

€

s = (10)π

2

⎛

⎝ ⎜

⎞

⎠ ⎟=

€

s = (10)π

5

⎛

⎝ ⎜

⎞

⎠ ⎟=

1.)

2.)

3.)

4.)€

s = rθ5.2cm

15.7cm

10.5cm

6.3cm

Find the degree measure of the central angle whose interceptedarc measures given, in a circle with radius 16 cm.

87

5.6

12

25

€

87 = (16)θ

€

5.6 = (16)θ

€

12 = (16)θ

€

25 = (16)θ

€

s = rθ

€

θ =87

16

€

θ =5.6

16

€

θ =12

16

€

θ =25

16

Now convertto degrees

Now convertto degrees

Now convertto degrees

Now convertto degrees

€

87

16•

180

π= 311.5°

€

5.6

16•

180

π= 20.1°

€

87

16•

180

π= 43°

€

87

16•

180

π= 89.5°

Find the area, to the nearest tenth, of the sector of a circle defined by a central angle given in radians, and the radius given.

€

θ =π6

,r =14

€

θ =7π

4,r =12

€

s =1

2r2θ

€

s =1

2(14)2 π

6

⎛

⎝ ⎜

⎞

⎠ ⎟= 51.3°

€

s =1

2(12)2 7π

4

⎛

⎝ ⎜

⎞

⎠ ⎟= 263.9°

Find the values of the six trig functions of an angle in standardposition if the point given lies on its terminal side.

(-1,5)

(6,-8)

(3,2)

(-3,-4)

θ

-1

5Use Pythagorean theorem to find the hypotenuse

€

26

€

sinθ =5 26

26

cscθ =26

5

€

cosθ =− 26

26

secθ = − 26

€

tanθ = −5

cotθ =−1

5

θ

6

-8

Use Pythagorean theorem to find the hypotenuse

€

10

€

sinθ =−4

5

cscθ =−5

4

€

cosθ =3

5

secθ =5

3

€

tanθ =−4

3

cotθ =−3

4

θ

3

2Use Pythagorean theorem to find the hypotenuse

€

13

€

sinθ =2 13

13

cscθ =13

2

€

cosθ =3 13

13

secθ =13

2

€

tanθ =2

3

cotθ =3

2

θ

-3

-4

Use Pythagorean theorem to find the hypotenuse

€

5

€

sinθ =−4

5

cscθ =−5

4

€

cosθ =−3

5

secθ =−5

3

€

tanθ =4

3

cotθ =3

4

Suppose θ is an angle in standard position whose terminal sidelies in the given quadrant. For each function, find the values of the remaining five trig functions of θ.

€

cosθ =3

5Quadrant I

€

sinθ =−2

3Quadrant IV

θ

3

4

Since we know cosine we can set up our triangle

€

5

€

sinθ =4

5

cscθ =5

4

€

cosθ =3

5

secθ =5

3

€

tanθ =4

3

cotθ =3

4

θ-2

Since we know sine we can set up our triangle

€

3

€

sinθ =−2

3

cscθ =−3

2

€

cosθ =5

3

secθ =3 5

5

€

tanθ =−2 5

5

cotθ =− 5

2

€

cosθ =adjacent

hypotenuse

Then use Pythagorean theorem to find the other leg

€

sinθ =opposite

hypotenuse

Then use Pythagorean theorem to find the other leg

€

5

Determine if the following are positive, negative,zero, or undefined.

€

sin11π

4

tanπ

2

sin(−45°)

cos450°

Quadrant IIPositive

Not in QuadUndefinded

Quadrant IVNegative

Not in QuadZero