This document discusses the unit circle and circular functions. It begins by explaining how the unit circle is used to define trigonometric function values and determine the measure of an angle based on its coordinates. It then defines the circular functions in terms of the unit circle and provides examples of evaluating circular function values both numerically and exactly. The document concludes by explaining linear and angular speed for a point rotating along a circle.

1. 6.2 Unit Circle and Circular
Functions
Chapter 6 Circular Functions and Their Graphs

2. Concepts and Objectives
 Use the unit circle to define values for trig functions.
 Determine the measure of an angle based on the
coordinates of its trig value.
 Determine linear and angular speed of a rotating point.

3. Unit Circle
 0 1,0
 

0,1
2
  1,0
 


3
0, 1
2
















x
y

4. Unit Circle
 0 1,0
 
 
 
2 2
,
4 2 2
 

0,1
2
 
 
 
3 2 2
,
4 2 2
  1,0
 
  
 
5 2 2
,
4 2 2
 


3
0, 1
2
 
 
 
7 2 2
,
4 2 2
















x
y

5. Unit Circle
 0 1,0
 
 
 
2 2
,
4 2 2
 
 
 
1 3
,
3 2 2
 

0,1
2
 
 
 
2 1 3
,
3 2 2
 
 
 
3 2 2
,
4 2 2
  1,0
 
  
 
5 2 2
,
4 2 2
 
  
 
4 1 3
,
3 2 2  


3
0, 1
2
 
 
 
5 1 3
,
3 2 2
 
 
 
7 2 2
,
4 2 2
















x
y

6. Unit Circle
 0 1,0
 
 
 
3 1
,
6 2 2
 
 
 
2 2
,
4 2 2
 
 
 
1 3
,
3 2 2
 

0,1
2
 
 
 
2 1 3
,
3 2 2
 
 
 
3 2 2
,
4 2 2
 
 
 
5 3 1
,
6 2 2
  1,0
 
  
 
7 3 1
,
6 2 2
 
  
 
5 2 2
,
4 2 2
 
  
 
4 1 3
,
3 2 2  


3
0, 1
2
 
 
 
5 1 3
,
3 2 2
 
 
 
7 2 2
,
4 2 2
 
 
 
11 3 1
,
6 2 2
















x
y

7. Circular Functions
 The circular functions of real numbers correspond to the
trigonometric functions of angles measured in radians.
r
s = 
x
y

(cos s, sin s) = (x, y)
Circular function values of
real numbers are obtained in
the same manner as
trigonometric function
values of angles measured in
radians.

8. Circular Functions (cont.)
For any real number s represented by a directed arc on
the unit circle,
sins y coss x  tan 0
y
s x
x
 
 
1
csc 0y
y
   
1
sec 0s x
x
   cot 0
x
s y
y
 

9. Circular Functions (cont.)
 Example: Find the exact values of and
7
cos
4
 
 
 
5
tan
3

10. Circular Functions (cont.)
 Example: Find the exact values of and
cos s = x, so the x-coordinate at
, and at , the coordinates are
7
cos
4
 
 
 
5
tan
3
7 2
4 2


tan
y
s
x


5
3
 
 
 
1 3
,
2 2

3
2
1
2
y
x

3
1
 3 or   
3 1 3 2
3
2 2 2 1

11. Calculator Tips
 I prefer to keep the calculators set in degree mode, so
what should we do when we need to keep switching
back and forth between degrees and radians?
 One solution might be to use the calculator for degrees
and set the scientific calculator on your phone to
radians.
 My preferred solution is to use a function on the
calculator that tells the problem your quantity is in
radians. Pressing /k, and then selecting r will add a
small r (almost like an exponent) to your number.

12. Approximating Circular Functions
 Example: Find a calculator approximation for each
circular function value.
(a) cos 1.85 (b) cot 1.3209 (c) sec(–2.9234)

13. Approximating Circular Functions
 Example: Find a calculator approximation for each
circular function value.
(a) cos 1.85 (b) cot 1.3209 (c) sec(–2.9234)
Make sure your values are in radians!
(a) cos 1.85 ≈ –.2756
(b) cot 1.3209 ≈ .2552
(c) sec(–2.9234) ≈ –1.0243

14. Approximating Circular Functions
 Example: Approximate the value of s in the interval
if cos s = .9685.
 
  
0,
2

15. Approximating Circular Functions
 Example: Approximate the value of s in the interval
if cos s = .9685.
cos–1 .9685 ≈ .2517
Since this value is in the quadrant given , this
is our value.
 
  
0,
2
1.57
2
 
 
 

16. Approximating Circular Values
 Example: Approximate the value of s in if
cos s = –.367.
 
  
3
,
2

17. Approximating Circular Values
 Example: Approximate the value of s in if
cos s = –.367.
cos–1 –.367 ≈ 1.947.
 
  
3
,
2
This angle is in QII, not QIII. To
find our angle, we need to
consider the angle with the same
x-value.
To find the “other” angle,
subtract the first angle from 2.
-.367

3
2
 2 1.947 4.337

18. Exact Circular Values
 Example: Find the exact value of s in the interval
if tan s = 1.
 
  
3
,
2

19. Exact Circular Values
 Example: Find the exact value of s in the interval
if tan s = 1.
tan s = 1 when x = y, which occurs at in the given
interval.
 
  
3
,
2
5
4

20. Linear and Angular Speed
 Suppose that point P moves at
a constant speed along a circle
of radius r. The measure of
how fast the position of P is
changing is called linear speed.
 If v represents linear speed,
then
r
s
x
y

P

distance
speed
time

s
v
t

21. Linear and Angular Speed
 As point P moves along the
circle, ray OP rotates around
the origin. The measure of how
fast POB is changing is called
angular speed.
 Angular speed, symbolized ,
is given as
where  is in radians.
r
s
x
y

P
O B

t



22. Linear and Angular Speed (cont.)
 Example: Suppose that point P is on circle O with radius
10 cm, and ray OP is rotating with angular speed /18
radians per second.
(a) Find the angle generated by P in 6 sec.
(b) Find the distance traveled by P in 6 sec.
(c) Find the linear speed of P in centimeters per second.

23. Linear and Angular Speed (cont.)
 Example: Suppose that point P is on circle O with radius
10 cm, and ray OP is rotating with angular speed /18
radians per second.
(a) Find the angle generated by P in 6 sec.


18
18 6
 

 
  
6
radians
18 3

24. Linear and Angular Speed (cont.)
 Example: Suppose that point P is on circle O with radius
10 cm, and ray OP is rotating with angular speed /18
radians per second.
(b) Find the distance traveled by P in 6 sec.
s r 
 
 
  
 
10
3
s


10
cm
3

25. Linear and Angular Speed (cont.)
 Example: Suppose that point P is on circle O with radius
10 cm, and ray OP is rotating with angular speed /18
radians per second.
(c) Find the linear speed of P in centimeters per second.

s
v
t


10
3
6
v
 
 
10 5
cm/sec
18 9

26. Classwork
 College Algebra
 Page 580: 8-34 (even), page 565: 54-60, 64-70 (even),
page 539: 32-56 (4)