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Precalculus Trigonometric Functions
The Unit Circle 2015
Precalculus WU 10/14
Find one positive and one negative angle
co-terminal with the given angle.
5

12
Objectives:
• Find trig function values for special angles using the
unit circle.
• Evaluate Trig functions using the unit circle.
• Use domain and period to evaluate trig functions.
• Solve application problems using the unit circle.
The six trigonometric functions of a right triangle, with an acute
angle , are defined by ratios of two sides of the triangle.
The sides of the right triangle are:
hyp
 the side opposite the acute angle ,
opp
 the side adjacent to the acute angle ,
θ
 and the hypotenuse of the right triangle.
adj
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
sin  =
cos  = adj
tan  = opp
hyp
hyp
adj
csc  =
hyp
opp
sec  = hyp
adj
Copyright © by Houghton Mifflin Company, Inc.
All rights reserved.
cot  = adj
opp
4
Trigonometric Functions
Let  be an angle in standard position with (x, y), a point on
the terminal side of  and r = x 2  y 2  0.
sin  
y
r
cos   x
r
y
y
, x0
x
cot   x , y  0
y
sec   r , x  0
x
csc   r , y  0
y
tan  
(x, y)

x
5
Example:
Determine the exact values of the six trigonometric functions
of the angle .
y
r  x 2  y 2  (3) 2  62
(3, 6)
 9  36  45

x
y
 6 2 5
r
5
45
cos   x  3  5
r
5
45
y
tan    6  2
x 3
sin  
csc   r  45  5
y
6
2
sec  r  45  5
x
3
cot   x  3  0.5
y 6
6
So, we know that trigonometric function
values are side length relationships of right
triangles.
We can easily evaluate the exact values of
trigonometric functions for special angles.
Geometry of the 30-60-90 triangle
Consider an equilateral triangle with
each side of length 2.
30○ 30○
The three sides are equal, so the
angles are equal; each is 60.
2
The perpendicular bisector
of the base bisects the
opposite angle.
60○
2
3
1
60○
2
1
Use the Pythagorean Theorem to
find the length of the altitude, 3 .
Copyright © by Houghton Mifflin Company, Inc.
All rights reserved.
8
Special right triangle relationships
2
3
2
1
45
60
1
1
Now, let’s apply it to the unit circle…
What does “unit circle” really mean?
It’s a circle with a radius of 1 unit.
What is the equation of the “unit circle”?
x  y 1
2
2

0,1
2
0, 0
1,0
2 , 360
 , 180
-1,0
0, -1
3
2
Let’s begin with an easy family…

4
What are the
coordinates?

2

4
1
 , 180

2
2
2
2
45
2
2
3
2
Now, reflect the triangle to the second quadrant…
0, 0
2 , 360
,
2
2

What are the
coordinates?

-
2
2
,
2
2


2

3
4
4
1
2
2
1
-
2
2
2
2
2
2
2
45
 , 180

2
3
2
Now, reflect the triangle to the third quadrant…
0, 0
2 , 360
,
2
2


2
-
2
,
2
2


2
3
4
4
1
2
2
-

-
2
2
, -
2
2

1
2
2
2
2
2
5
4

2
2
2
45
 , 180
What are the
coordinates?

3
2
Now, reflect the triangle to the fourth quadrant…
0, 0
2 , 360
,
2
2


2
-
2
,
2
2


2
3
4
4
1
2
2
-

-
2
, -
2
2

1
7
3
2
,
2
2
2

0, 0
2 , 360
2
2
5
4
2
2
2

2
2
45
 , 180
2

4

What are the
coordinates?
2
2
, -
2
2

Complete the

family…
6
.

6

1
30
3
2
,
1
2

1
2
3
2
Now, reflect the triangle to the second quadrant.

3 1
,
2 2


6

5
6
1
2
-
3
2
1
1
30
2
3
2
,
1
2
3
2
Now, reflect the triangle to the third quadrant.


3 1
,
2 2

5
6
1
2
-
What are the
coordinates?

-
3
2
, -
1
2


6

7
3
2
1
1
30
2
3
2
,
3
2
6
Now, reflect the triangle to the fourth quadrant.
1
2


3 1
,
2 2

5
6
1
2
-

-
3
2
, -
1
2


6

7
6
3
2
1
1
30
2
3
2
,
1
2

3
2
11

3
2
, -
1
2

6 What are the
coordinates?
Let’s look at another “family”

3



2
3
2
,
3
2

3
1
2
 , 180
1
60
1
2
3
2
Now, reflect the triangle to the second quadrant
0, 0
2 , 360
What are the
coordinates?

-
1
2
,


3
2


2
2
3
3
1
3
2
-
1
60
1
2
2
,
3
2

3
1
2
 , 180
1
2
3
2
Now, reflect the triangle to the third quadrant
0, 0
2 , 360



-
1
2
3
,
2

2
3
1
2
-
1
60
1
2
2
What are the
coordinates?

-
2
, -
2

2
,
3
2

4
3
3
1
2
 , 180
3
1
3
3
1

2
3
2
Now, reflect the triangle to the fourth quadrant
0, 0
2 , 360



-
1
2
3
,
2

2
3
1
2
-

-
2
, -
2

2
,
3
2

4
3
3
1
2
 , 180
3
1
3
3
1

2
1
0, 0
2 , 360
60
1
2
2
3
2
5
3

What are the
coordinates?
1
2
, -
3
2

Ordered pairs of special angles around the Unit Circle
 1 3
 2, 2 


Since r = 1…
cos   x
1

3,1

 2 2



(–1, 0)

3 1
 2 , 2 



2
 2 ,

 x, y    cos ,sin  
90°
2
3 3 120°
5 4 135°
6 150°

2 2
 2 , 2 


y
sin  
1
y
180°
(0, 1)

2

60° 3
1 3
2, 2 


 2 2
 2 , 2 



4
45°

30° 6
 3 1
 2 ,2


0° 0 x
360° 2 (1, 0)
330°
11
315°
 3 1
7 210°
6
6  2 , 2 
225°


7
5
240° 300°
2  4 4
5 4  2 ,  2 

2 
2 
 2
3 270° 32 3
 1
3

,

 2
2 

(0, –1)
1
3
,

2
2 

24
• Important point:
Since r = 1…
sin  
y
1
cos   x
1
 x, y    cos ,sin  
Because ordered pairs around the unit circle (x, y) represent
2
2
x

y
1,
sine and cosine, and the equation of the circle is
We have the following identity: cos 2   sin 2   1
What if the radius is not 1?
6
1
30
30
Trigonometric values are functions of the angle – ratios of
sides of similar triangles remain the same. So it always holds
that cos 2   sin 2   1.
Trigonometric Values
of Special Angles
 1 3
 2, 2 



(–1, 0)

3 1
 2 , 2 



2
 2 ,

90°
2
3 3 120°
5 4 135°
6 150°

2 2
 2 , 2 



3,1

 2 2


y
180°
(0, 1)

2

60° 3
1 3
2, 2 


 2 2
 2 , 2 



4
45°

30° 6
 3 1
 2 ,2


0° 0 x
360° 2 (1, 0)
330°
11
315°
 3 1
7 210°
6
6  2 , 2 
225°


7
5
240° 300°
2  4 4
5 4  2 ,  2 

2 
2 
 2
3 270° 32 3
 1
3

,

 2
2 

(0, –1)
1
3
,

2
2 

27
Domain
The domain of the sine and cosine function is the set of all real
numbers.
(0, 1) y
Unit Circle
(–1, 0)
(1, 0)
x
1  y  1
(0, –1)
1  x  1
Range
The point (x, y) is on the unit circle, therefore the range of the
sine and cosine function is between – 1 and 1 inclusive.
28
A function f is periodic if there is a positive real number
c such that
f (t + c) = f (t)
for all t in the domain of f. The least number c for which
f is periodic is called the period of f.
y
Unit Circle
f (t)  sin t
Periodic Function
x
t = 0, 2, …
Period
29
Example:
Evaluate sin 5 using its period.
5 - 2 - 2 = 
sin 5 = sin  = 0
f (t)  sin t
y
(–1, 0)
x
Adding 2 to each value of t in the
interval [0, 2] completes another
revolution around the unit circle.
sin(t  2 n)  sin t
cos(t  2 n)  cos t
30
You Try:
9
a) Evaluate sin
4
b) Evaluate cos
10
3
Can you?
Evaluate each of the following. Exact values only please.
5
tan
6
3
sec
4
Even and Odd Trig Functions
Remember:
if f(-t) = f(t) the function is even
if f(-t) = - f(t) the function is odd
The cosine and secant functions are EVEN.
cos(-t)=cos t
sec(-t)=sec t
(0,1) y
(–1, 0)
(0,–1)
The sine, cosecant, tangent, and cotangent functions are
ODD.
sin(-t)= -sin t
csc(-t)= -csc t
tan(-t)= -tan t
cot(-t)= -cot t
(1, 0)
Example:
Evaluate the six trigonometric functions at  = .
sin   y  0
cos   x  1
y 0
tan   
0
x 1
csc   1  1 is undefined.
y 0
sec   1  1  1
x 1
cot   x  1 is undefined.
y 0
(0, 1) y

(–1, 0)
x
(1, 0)
(0, –1)
34
Application:
A ladder 20 feet long leans against the side of a house.
The angle of elevation of the ladder is 60 degrees.
Find the height from the top of the ladder to the ground.
Application:
An airplane flies at an altitude of 6 miles toward a point
directly over an observer. If the angle of elevation
from the observer to the plane is 45 degrees, find the
horizontal distance between the observer and the plane.
.
Homework
4.2 pg. 264
1-51 odd
Trig Races
4
cos
3
1

2
 3
3
tan210
11
csc
4
2
csc7 undef.
22
sin

3

cos
3
3
2
1
2
7
sin
3
3
sin
2
4
sec
3
3
2
3
csc
4
1
7
tan
3
2
8
cot
3
 2
 3
3

3
HWQ 10/15
Evaluate each of the following. Exact values only please.
5
tan
6
3
sec
4
csc7
22
sin
3