Download 08-General - WVU Math Department

y

r
(a, b)
x
Let  be any angle in standard position, and
let a , b denote the coordinates of any point,
except the origin (0, 0), on the terminal side
of  . If r  a  b denotes the distance from
2
2
(0, 0) to (a , b), then the six trigonometric
functions of  are defined as the ratios
sin   b r
cos  a r
tan   b a
csc  r b
sec  r a
cot   a b
provided no denominator equals 0.
Find the exact value of each of the six
trigonometric functions of a positive angle 
if (-2, 3) is a point on the terminal side.
(-2, 3)
y

x
a  2, b  3
r  a  b  ( 2)  3  13
2
2
b
3
3 13
sin   

r
13
13
2
2
r
13
csc  
b
3
a  2  2 13
r
13
cos  

sec   
r
13
13
a
2
b 3
3
tan   

a 2
2
a
2
cot    
b
3
b 0
sin 0  sin 0    0
r
1
y
a 1

cos 0  cos 0    1
r 1
b 0

tan 0  tan 0    0
a 1
r 1
r 1


csc
0

csc
0


x
b 0
P= (1, 0) sec 0  sec 0  r  1  1
a 1
P= (a, b)
a 1

cot 0  cot 0  
b 0


90o ?
a=0
b=1
r=1
b 1

sin  sin 90    1
2
r 1

a 0

cos  cos 90    0
2
r 1

b 1

tan  tan 90  
2
a 0

r 1

csc  csc 90    1
2
b 1

r 1

sec  sec 90  
2
a 0

a 0

cot  cot 90    0
2
b 1
180 ( radians)

270 (3 2 radians)

sin 
0
1
cos
1
0
tan 
0
Not defined
csc 
Not defined
1
sec 
1
Not defined
cot 
Not defined
0
y
a < 0, b > 0, r > 0
a > 0, b > 0, r > 0

r
x
(a, b)
a < 0, b < 0, r > 0
a > 0, b < 0, r > 0
y
II  ,  
sin   0, csc   0
I (+, +)
All positive
All others negative
III  , 
IV  , 
tan   0, cot   0
cos  0, sec   0
All others negative
All others negative
x
Let  denote a nonacute angle that
lies in a quadrant. The acute angle
formed by the terminal side of  and
either the positive x-axis or the
negative x-axis is called the reference
angle for  .
Reference
Angle

y
  180  
  

 
x
    180
 

  360  
 2  

Find the exact value of each of the
following trigonometric functions using
reference angles:
16

(a) cos 570
(b) tan
3
(a) 570  360  210  



 in Quadrant III, so cos < 0
  210  180  30



3
cos 210   cos 30  
2


16
16 6 10
 2 


 b
3
3
3
3
10 6 4


3
3
3
 is in Quadrant III, so tan > 0
4


 
3
3
16

3
tan
 tan 
3
3 2