5.3 Trigonometric Functions of Any Angles.
The Unit Circle
1.
2.
3.
4.
Use the definitions of trigonometric functions of any angle.
Use the signs of the trigonometric functions.
Find reference angles.
Use reference angles to evaluate trigonometric functions.
Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu
H.Melikian/1200
1
Definitions of Trigonometric Functions of Any Angle

Let  be any angle in standard position and let P = (x, y) be
a point on the terminal side of  If r  x 2  y 2 is the
distance from (0, 0) to (x, y), the six trigonometric functions
of  are defined by the following ratios:
H.Melikian/1200
y
sin  
r
r
csc  , y  0
y
x
cos 
r
r
sec  , x  0
x
y
tan   , x  0
x
x
cot   , y  0
y
2
Example: Evaluating Trigonometric Functions
Let P = (1, –3) be a point on the terminal side of  Find each of
the six trigonometric functions of 
P = (1, –3) is a point on the terminal side of 
x = 1 and y = –3
r  x 2  y 2  (1)2  (3)2  1  9  10
y 3 3
sin   

r
10 10
1
x

cos  
r
10
H.Melikian/1200
10
3 10

10
10
1
10
10

10 10 10
3
Example: Evaluating Trigonometric Functions
(continued)


Let P = (1, –3) be a point on the terminal side of
of the six trigonometric functions of 
We have found that r  10.
y 3
 3
tan   
x 1
r
csc    10
y
3
 Find each
1
x
cot    
3
y
r
10
sec  
 10
x
1
H.Melikian/1200
4
Example: Evaluating Trigonometric Functions
(continued)

Let P = (1, –3) be a point on the terminal side of
of the six trigonometric functions of

Find each

H.Melikian/1200
3 10
sin   
10
10
csc  
3
10
cos 
10
sec  10
tan   3
1
cot   
3
5
Example: Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the cosine function and the cosecant function at
the following quadrantal angle:
If   0  0
then the terminal side of the angle is on the positive
x-axis. Let us select the point P = (1, 0) with x = 1 and y = 0.
  0  0 radians,
x 1
cos    1
r
1
r
1
csc  
y
0
csc
H.Melikian/1200
is undefined.
6
Example: Trigonometric Functions of Quadrantal Angles


Evaluate, if possible, the cosine function and the cosecant function at the
following quadrantal angle:
If
  90 

2
then the terminal side of the angle is on the
positive y-axis. Let us select the point P = (0, 1) with x = 0 and y = 1.
  90 

2
radians,
x 0
cos    0
r 1
r 1
csc    1
y 1
H.Melikian/1200
7
Example: Trigonometric Functions of
Quadrantal Angles


Evaluate, if possible, the cosine function and the cosecant
function at the following quadrantal angle:   180  
If   180   radians, then the terminal side of the angle is
on the positive x-axis. Let us select the point P = (–1, 0)
with x = –1 and y = 0.
x 1
cos    1
r 1
r
1
csc  
y
0
csc
H.Melikian/1200
is undefined.
8
Example: Trigonometric Functions of Quadrantal Angles


Evaluate, if possible, the cosine function and the cosecant
3
function at the following quadrantal angle:   270 
2
3
radians, then the terminal side of the angle
If   270 
2
is on the negative y-axis. Let us select the point P = (0, –1)
with x = 0 and y = –1.
x 0
cos    0
r 1
r
csc   1  1
y 1
H.Melikian/1200
9
The Signs of the Trigonometric Functions
H.Melikian/1200
10
Example: Finding the Quadrant in Which an Angle Lies

If sin   and cos  0, name the quadrant in which the
angle  lies.
 lies in Quadrant III.
H.Melikian/1200
11
Example: Evaluating Trigonometric Functions


Given tan    1 and cos  0, find sin  and sec .
3
Because both the tangent and the cosine are negative, 
lies in Quadrant II.
y
1
tan   
x 3
x  3, y  1
r  x 2  y 2  (3)2  (1)2  9  1  10
y
1
sin   
r
10
H.Melikian/1200
10
10

10 10
10
10
r

sec  
3
x 3
12
Definition of a Reference Angle
H.Melikian/1200
13
Example: Finding Reference Angles



Find the reference angle,   for each of the following
angles:
a.   210      180  210  180  30
b.   7
4
   2    2  7  8  7  
4
4

c.
  240    60

d.
  3.6        3.6  3.14  0.46
H.Melikian/1200
4
4
14
Finding Reference Angles for Angles Greater Than 360°
(2 ) or Less Than –360° ( 2 )
H.Melikian/1200
15
Example: Finding Reference Angles

Find the reference angle for each of the following angles:

a.


  665
   360  305  55
15
7 8 7 
b.  
   2 



4
c.    11
3
H.Melikian/1200
4
4
4
11 12 
  


3
3
3
4
16
Using Reference Angles to Evaluate Trigonometric Functions
A Procedure for using reference Angles to Evaluate Trigonometric Functions
H.Melikian/1200
17
Example: Using Reference Angles to Evaluate Trigonometric
Functions
sin135.

Use reference angles to find the exact value of

Step 1 Find the reference angle,  and sin 
   360    360  300  60

Step 2 Use the quadrant in which  lies to prefix the appropriate sign
to the function value in step 1.
3
sin 300   sin 60  
2
H.Melikian/1200
18
Example: Using Reference Angles to Evaluate Trigonometric
Functions


5
Use reference angles to find the exact value of
tan
.
4
Step 1 Find the reference angle,  and tan 
5 4 

   


4

4
4
Step 2 Use the quadrant in which  lies to prefix the appropriate sign
to the function value in step 1.
tan
H.Melikian/1200
5
 1
 tan
4
4
19
Example: Using Reference Angles to Evaluate Trigonometric
Functions

   .
sec
Use reference angles to find the exact value of


 6

Step 1 Find the reference angle,   and sec .

Step 2 Use the quadrant in which  lies to prefix the
appropriate sign to the function value in step 1.


sec 


sec


6
6


H.Melikian/1200
2 3

3
20