6.2 Unit Circle and Circular Functions

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

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.

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

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

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

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

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.

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
 

 Example: Find the exact values of and
7
cos
4
 
 
 
5
tan
3

 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

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.

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)

 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

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

 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
 
 
 

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

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

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

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

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

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



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.

radians per second.
(a) Find the angle generated by P in 6 sec.


18
18 6
 

 
  
6
radians
18 3

radians per second.
(b) Find the distance traveled by P in 6 sec.
s r 
 
 
  
 
10
3
s


10
cm
3

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

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

6.2 Unit Circle and Circular Functions

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