Download Trigonometric Functions: The Unit Circle

Sullivan Precalculus:
Section 5.2
Trig Functions: Unit Circle
Objectives of this Section
• Find the Exact Value of the Trigonometric Functions Using
the Unit Circle
•Find the Exact Value of the Trigonometric Functions of 45
Degrees
• Find the Exact Value of the Trigonometric Functions of 30
Degrees and 60 Degrees
• Use a Calculator to Approximate the Value of the
Trigonometric Functions of Acute Angles
The unit circle is a circle whose radius
is 1 and whose center is at the origin.
Since r = 1:
s  r
becomes
s
(0, 1) y
s 

x
(1, 0)
(-1, 0)
(0, -1)
(0, 1) y
 t
P = (a, b)

(-1, 0)
(1, 0)
(0, -1)
x
Let t be a real number and let P = (a, b) be
the point on the unit circle that corresponds
to t.
The sine function associates with t the
y-coordinate of P and is denoted by
sint  b
The cosine function associates with t the
x-coordinate of P and is denoted by
cost  a
If a  0 , the tangent function is defined as
b
tan t 
a
If b  0 , the cosecant function is defined as
1
csc t 
b
If a  0 , the secant function is defined as
1
sect 
a
If b  0 , the cotangent function is defined as
a
cot t 
b
1
15 
 be
Let t be a real number and let P   ,

4
4


the point on the unit circle that corresponds to t.
Find the exact value of the six trigon ometric
functions.

a , b    1 4 ,

15
sint  b  
4
15

4 
1
cost  a 
4
a , b 
  1 , 
 4
15

b
4 
tan t  
1
a
4
1
1
csct  

b  15
4
1 1
sect  
4
a 1
4
1
a
4 
cot t  
b  15
4
15

4 
15
4
4 15

15
15
1
15

15
15
(0, 1) y
 t
P = (a, b)

(-1, 0)
(1, 0)
(0, -1)
x
If   t radians, the six trigonometric
functions of the angle  are
defined as
sin   sin t
cos  cos t
tan   tan t
csc  csc t
sec  sec t
cot   cot t
Find the exact value of the six trigonometric
functions of 45 degrees.
45
c
b=1
2
45
a=1
c  a b
2
2
2
c 1 1  2
2
c
2
2
2

b 1
2
sin 45  sin  

4 c
2 2
 a 1
2
cos 45  cos  

4 c
2 2
 b 1
 a 1
tan45  tan    1 cot 45  cot    1
4 a 1
4 b 1
 c
2
csc 45  csc  
 2
4 b 1
 c
2
sec 45  sec  
 2
4 a
1
Find the exact value of the six trigonometric
functions of 30 and 60 degrees.
30 30
2
2
b
60
60
a
a
2 1 b
2
2
2
b  2 1  4 1 3
b 3
2
2a = 2 so a = 1
c  a b
2
2
2
2
2
 a 1
 b
3
sin 30  sin   cos30  cos  
6 c 2
6 c 2
 sin 30
12
1
3
tan 30  tan 



6 cos 30
32
3 3

1
1
cot 30  cot 

 3
6 tan 30 1 3

1
1
csc 30  csc 

2
6 sin 30 1 2

1
1
2 2 3
sec 30  sec 



6 cos 30
32
3
3
 a 1
3
cos60  cos  
sin60  sin  
3 c 2
3 c 2
 sin 60
32
3
tan 60  tan 


 3
3 cos60
12
1

1
1
3
cot 60  cot 


3 tan 60
3 3

1
1
2 2 3
csc 60  csc 



3 sin 60
32
3
3

1
1
sec 60  sec 

2
3 cos60 1 2
 b
Use a calculator to find the approximate
value of
(a) sin52
( b) tan

5
(c) sec

5