1. Review
Radian Measure and
Circular Functions
Rev.S08 1

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. 2 2

3. Example of Converting from Degrees to Radians
Convert each degree measure to radians.
a) 60
b)  OR 3

4. Example of Converting from Radians to Degrees
Convert each radian measure to degrees.
a)‏
b) 3.25 4

5. Let’s Look at Some Equivalent Angles in Degrees and Radians
6.28 2π
360
1.05
60
4.71
270
.79
45
3.14 π
180
.52
30
1.57
90 0
Approximate
Exact
Radians
Degrees
Rev.S08 5 5

6. Let’s Look at Some Equivalent Angles in Degrees and Radians (cont.)‏6 6

7. Examples Find each function value. a)‏ b)
IF YOUR ANGLE IS ON THE UNIT CIRLE THEN USE IT!!
Or you can set up your triangles!
tan = opp
adj
sin = opp
hyp θ θ 1 1 7 7

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. 8 8

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 9 9

10. Example of Application
A rope is being wound around a drum with radius ft. How much rope will be wound around the drum it the drum is rotated through an angle of 39.72?
Convert to radian measure. 10 10

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? 11 11

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. 12 12

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. 13 13

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 14 14

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. 15

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. 16 16

17. Circular Functions and their Reciprocals
This is an example of a triangle in the 1st quadrant y 1 y θ y y x 17 17

18. 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. 19 19

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 20

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

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

23. Change the given radian measure into degrees.
=-57.3°
=720°
=33.75°
=-140°

24. Find one positive and one negative angle that is coterminal with
an angle measuring the given θ
add 360°
subtract 360°
--290°, 430°
add 2π
subtract 2π
add 360°
subtract 360°
--660°, 60°
add 2π
subtract 2π

25. Is the acute angle formed with the x-axis
Find the reference angle for the angle given:
Is the acute angle formed with the x-axis
20° θ θ
20°
one full revolution
With left over θ θ
-112.5°

26. 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.)
3.)
5.2 cm
15.7 cm
4.)
2.)
6.3 cm
10.5 cm

27. Find the degree measure of the central angle whose intercepted
arc measures given, in a circle with radius 16 cm. 87
5.6 12 25
Now convert
to degrees
Now convert
to degrees
Now convert
to degrees
Now convert
to degrees

28. 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.

29. Find the values of the six trig functions of an angle in standard
position if the point given lies on its terminal side.
Use Pythagorean theorem to find the hypotenuse 5
(-1,5)
(6,-8)
(3,2)
(-3,-4) θ -1
Use Pythagorean theorem to find the hypotenuse 6 θ -8
Use Pythagorean theorem to find the hypotenuse 2 θ 3
Use Pythagorean theorem to find the hypotenuse -3 θ -4

30. 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.
Quadrant I
Since we know cosine we can set up our triangle 4 θ 3
Then use Pythagorean theorem to find the other leg
Quadrant IV
Since we know sine we can set up our triangle θ
Then use Pythagorean theorem to find the other leg -2

31. Quadrant II Positive Not in Quad Undefinded Quadrant IV Negative
Determine if the following are positive, negative,zero, or undefined.
Quadrant II
Positive
Not in Quad
Undefinded
Quadrant IV
Negative
Not in Quad
Zero