Review 5.1-5.3
Radian Measure and
Circular Functions
Rev.S08
How to Convert Between Degrees and
Radians?

1. Multiply a degree measure by
simplify to convert to radians.

2. Multiply a radian measure by
to convert to degrees.
radian and
and simplify
2
Example of Converting from
Degrees to Radians

Convert each degree measure to radians.
a)
60

b) 221.7

OR
739
600
3
Example of Converting from
Radians to Degrees

Convert each radian measure to degrees.

a)

b) 3.25
4
Let’s Look at Some Equivalent Angles in
Degrees and Radians
Degrees
Radians
Exact
0
Approximate
Radians
Exact
0
90
30
.52
180
45
.79
270
60
1.05
360
Rev.S08
0
Degrees
Approximate
1.57

3.14
4.71
2
6.28
5
Let’s Look at Some Equivalent Angles in
Degrees and Radians (cont.)
6

Examples


Find each function value.
a)

b)
IF YOUR ANGLE IS ON THE UNIT CIRLE THEN USE IT!!
 3
y
2   3  2 
tan  

x
2
1
1
 3

2
3
2
Or you can set up your triangles!

tan = opp
1
2
 3
 2

adj
 3
2

θ
1

1
2
θ
sin = opp
hyp
1
7
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  radians is
given by the product of
the radius and the radian
measure of the angle, or
s = r ,  in radians.
8
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
9
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.
10
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?
11
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.
12
Let’s Practice Another Application of
Radian Measure Problem (cont.)

An arc with this length on the larger gear
corresponds to an angle measure  , in radians
where

Convert back to degrees.
13
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  is given by
14
Example



Find the area of a sector with radius 12.7 cm and
angle  = 74.
Convert 74 to radians.
Use the formula to find the area of the sector of a
circle.
15
What is a Unit Circle?

A unit circle has its center at the origin and a
radius of 1 unit.
Note: r = 1
s
= r ,
s= in radians.
16
Circular Functions and their Reciprocals
This is an example of
a triangle in the 1st
quadrant
1
y
y
y
θ
x
y
17
Remember our two special triangles
that make up the unit cirlce:
Let’s Look at the Unit Circle Again
Because its made up of our “special”
triangles.
19
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

20
IF AN ANGLE IN STANDARD POSITION MEASURES
THE GIVEN RADIANS, DETERMINE WHICH
QUADRANT IT’S TERMINAL SIDE LIES.
7
12
II
2
3
III
371
I
14
5
II
Change the given degree measure to radian measure
in terms of π.
36
250

145

6


3

10

180


 180


180


180
25

18
29

36


30
Change the given radian measure into degrees.
1
4

3
16

7
9



180

=-57.3°
180


=720°
180


=33.75°
180

=-140°
Find one positive and one negative angle that is coterminal with
an angle measuring the given θ
70
2
5
300
3
4
add 360°
subtract 360°
add 2π
subtract 2π
add 360°
subtract 360° 
add 2π
subtract 2π
--290°, 430°
8 12
,
5
5
--660°, 60°
11 5
,
4
4
Find the reference angle for the angle given:
20
160
10
3
5
8
θ
θ
Is the acute
angle formed
with the x-axis
20°
one full revolution
With 4  left over
3
θ

θ
20°

-112.5°
4  3 


3
3
3
5

8
8 5 3


8
8
8
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
1.) 
6
 
s  (10) 
6 
5.2
cm

90
 
s  (10) 
2 
15.7
cm

4.) 36
2.) 
3
3.)
s  r

 
s  (10) 
3 
10.5
cm
 
s  (10) 
5 
6.3
cm



Find the degree measure of the central angle whose intercepted
arc measures given, in a circle with radius 16 cm.
87
5.6
87  (16)
87

16
5.6  (16)
 
5.6

16

12
 
12  (16)
12

16
25
25  (16)
 
25

16
s  r
Now convert
to degrees
Now convert

to degrees

Now convert
to degrees
Now
convert
to degrees
87 180

 311.5
16 
5.6 180

 20.1
16

87 180

 43
16 
87 180

 89.5
16 
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.

  ,r  14
6
7


4
,r  12
 
1
2 
s  (14)   51.3
6 
2

 
1
2 7
s  (12)   263.9
 4 
2
1 2
s r 
2
Find the values of the six trig functions of an angle in standard
position if the point given lies on its terminal side.
26
(-1,5)
5

Use Pythagorean
theorem to find the
hypotenuse
θ
-1
theorem to find the
hypotenuse
6
θ
-8
10
13
(3,2)
θ
3

-3
(-3,-4)


Use Pythagorean
(6,-8)
θ
-4
5
2
Use Pythagorean
the
theorem to find
hypotenuse
5 26
26
26
csc  
5
sin  
4
5
5
csc  
4
sin  
2 13
13
13
csc  
2
sin  
Use Pythagorean
theorem to find the
hypotenuse

4
5

5
csc  
4
sin  
 26 tan   5
26
1
cot  
5
sec    26
cos  

3
5
5
sec  
3
cos  
3 13
13
13
sec  
2
cos 
3
cos  
5

5
sec  
3
4
3
3
cot  
4
tan  
2
3
3
cot  
2
tan  
4
3
3
cot  
4
tan  
Suppose θ is an angle in standard position whose terminal side
lies in the given quadrant. For each function, find the values
of the remaining five trig functions of θ.
3
cos 
5
cos  
Quadrant I
5

2
sin  
3
4

θ
3
adjacent
hypotenuse
Since we know
cosine we can set
up our triangle
Then use
Pythagorean
theorem to find the
other leg
Quadrant IV

Since we know sine
we can set up our
triangle
5

θ


3
-2
Then use
Pythagorean
theorem to find the
other leg

4
5
5
csc  
4
sin  
sin 

2
3
3
csc  
2
sin  
3
5
5
sec  
3
cos  
4
3
3
cot  
4
tan  
opposite

hypotenuse
5
3
3 5
sec  
5
cos 
2 5
5
 5
cot  
2
tan  
Determine if the following are positive, negative,zero, or undefined.
11
sin
4
Quadrant II
Positive
 Not in Quad
tan
2 Undefinded
sin( 45)Quadrant IV
Negative
Not in Quad
cos450 Zero