Download Inverse Trigonometric Functions

Trigonometric
Functions
Section 1.6
Radian Measure
• The radian measure of the angle ACB at the center of
the unit circle equals the length of the arc that ACB
cuts from the unit circle.
Radian Measure
• An angle of measure θ is placed in standard position
at the center of circle of radius r,
In this course, you
will see that so
much of calculus
only works when
measuring angles
in radians!!!
Trigonometric Functions of Theta
The six basic trigonometric functions of  are
defined as follows:
y
sine: sin  
r
x
cosine: cos  
r
y
tangent: tan  
x
r
cosecant: csc  
y
r
secant: sec  
x
x
cotangent: cot  
y
 1 3
  ,

 2 2 

2 2
,
 

2
2


3

2
2
3
120

3 1
,  5 4
 
135
 2 2
6 150
(–1,0) 180
1 3
 ,

2
2

90  
 2 2
,


3
2
2



60
45 4   3 1 
, 

 2 2
(0,1)
30 6
0 0 (1,0)

360 2
210
33011 
7

3 1

3 1
,   6
315 6  2 ,  2 
 
225
2


 2
7
5
300
240


 2

2
2
2
5

4
,
,
4 4
 



3

2 
2 
 2
 2
3
3 270
 1

1
3
3
2
 ,

 ,

 2

2 
(0,–1)
2

2 
Other Assortments…
A function f  x  is periodic if there is a positive number p such
that f  x  p   f  x  for every value of x. The smallest value
of p is the period of f .
The functions cos x, sin x, sec x and csc x are periodic with
period 2 . The functions tan x and cot x are periodic with
period  .
• The graphs of cos x and sec x are even functions
because their graphs are symmetric about the y-axis.
• The graphs of sin x, csc x, tan x and cot x are odd
functions.
Inverse Trigonometric Functions
• None of the six basic trigonometric functions graphed
in Figure 1.42 is one-to-one. These functions do not
have inverses. However, in each case, the domain can
be restricted to produce a new function that does have
an inverse.
• The domains and ranges of the inverse trigonometric
functions become part of their definitions.
Inverse Trigonometric Functions
Function
Domain
Range
y  cos1 x
1 x1
0 y 
y  sin x
1
1
y  tan x
1
y  sec x
1
1 x1
  x  

  y
2
2

x 1
y  cot 1 x
  x  

2
 y

2
0  y  , y 
x 1
y  csc x



 y

2
2
0 y 

2
,y0
Inverse Trigonometric Functions
A key to remember:
Inverse trig functions ARE angles!!!
How to “read” inverse trig expressions…
sin
1
 x
“The sine (or other
trig function) of
what angle is x?”
Also remember: Inverse sine and inverse tangent are
always on the right half of the unit circle, and inverse
cosine is always on the top half of the unit circle…
Guided Practice
Evaluate each of the following:
5 1
sin

6
2
 3
cot  
 4

  1

13
3
tan

6
3
 21
sec  
 4

 2

1
4

cos
2
3

2 3
csc 
3
3
Guided Practice
Determine (a) the period, (b) the domain, and (c) the
range, and (d) draw the graph of the function.
y  2sin  4 x     3  2sin  4  x   4    3
Horizontal shift (phase shift) left  4, horizontal shrink
by 1/4, vertical stretch by 2, vertical shift up 3
2 2 


(a) Period:
b
4
2
 ,  
(c) Range: 1,5
(d) How does the graph look?
(b) Domain:
Guided Practice
Solve the equation in the specific interval.
cos x  0.6, 0  x  3
x  0.927,5.356, 7.210
3
sin x 
,
2
  x  

2
x   2n ,
 2n
3
3
n is any integer
Guided Practice
Evaluate each of the following:


2
sin  
  
4
 2 
1
cos
1
 0.23  1.339
 1  1  

3

cot  cos     cot   
3
 2 

3
9
 1  9  
 0.959
tan  sin    
 13   2 22

Let’s complete this last one with an analytic graph…