Download Sec 2.2

Chapter 2
Trigonometric Functions of Real Numbers
Section 2.2
Trigonometric Functions of Real Numbers
Trigonometric Functions
If t is a real number that is the length of an arc on the unit circle and P is a point at
the end of that arc with coordinates (x,y) then we get the following expressions for
each of the trigonometric ratios:
y
sin t  y
cos t  x
tan t  ( x  0)
x
1
1
x
csc t  ( y  0) sec t  ( x  0) cot t  ( y  0)
y
x
y
P(x,y)
t
1
Find the values of the trigonometric functions at t = /6.
sin

6

1
2
csc 6  2
cos 6 
sec 6 
3
2
2 3
3
tan 6 
cot 6 
1
2
3
2
3
2
1
2

1
3

3
3
t  6
P
1
 3

3
2
, 12

Find the values of the trigonometric functions at t = 3/2.
sin
3
2
 1
csc 32  1
cos 32  0
tan 32 undefined
sec 32 undefined
cot 32  0
t
1
3
2
P0,1
This concept for the trigonometric functions
agrees with what we learned about the
trigonometric ratios. This is a special case
where the point on the terminal side of the
angle in standard position is on the unit
circle. Notice x is the length of the adjacent
side and y is the length of the opposite side
and the hypotenuse is of length 1.
opposite
y
sin t 
 y
hypotenuse 1
adjacent
x
cos t 
 x
hypotenuse 1
P ( x, y )
1
opposite
adjacent
1
t  2 , P(0,1)

t  3 , P 12 ,
The values of the trigonometric functions will be
the same as that of the trigonometric ratios. In
particular for the angles 0,/6,/4,/3,/2 (i.e.
0°,30°,45°,60°,90°). In the picture to the right the
first quadrant is shown along with the terminal
points on the unit circle.
3
2

t  4 , P

2
2
2
2
,

t  6 , P
3
2

, 12
t  0, P (1,0)
sin t cos t tan t cot t sec t csc t
0
0
1
0

6
1
2
3
2
3
3

4

3
2
2
2
2
1
3
2
1
2
1
0

2
3
-
-
1
-
3
2 3
3
2
1
2
2
3
3
2
2 3
3
0
-
1

Reciprocal Identities:
1
csc t 
sin t
1
sec t 
cos t
1
cot t 
tan t
Even & Odd Properties
Changing the direction of the angle from clockwise to counterclockwise or visa
versa makes no difference to the cosine and secant, they are called even
functions. The sine, tangent, cotangent and cosecant a change in direction
makes them the negative of what they were.
t , P ( x, y )
sin( t )   sin t
csc( t )   csc t
cos( t )  cos t
sec( t )  sec t
tan( t )   tan t
cot( t )   cot t
 t , P ( x,  y )
Pythagorean Identities:
sin 2 t  cos 2 t  1 tan 2 t  1  sec 2 t 1  cot 2 t  csc 2 t
These can be very useful when trying to do the following problem.
If t is an angle in the second quadrant and sin t = 5/6 find the other trigonometric
functions of t.
5
6
 11
cos t 
6
sin t
tan t 

cos t
sin t 
2
5
2
   cos t  1
6
25
 cos 2 t  1
36
11
2
cos t 
36
11
cos t  
6
5
6
 11
6
5
 5 11

11
 11
1
 11

tan t
5
1
1
 6 11
sec t 
  11 
cos t
11
6
cot t 
second
quadrant cos
is negative

csc t 
1
6

sin t 5