Since the trig functions are all periodic graphs, none of them
pass the horizontal line test. Thus, none of the graphs are 1-1 and
do not have inverse functions. What we can do is restrict the domain
of each of the trig functions to make each one, 1-1. Since the graphs
are periodic, if we pick an appropriate domain, we can use all values for
the range.
If we use the domain: -p/2 <
x < p/2, we have made the graph 1-1.
Notice, every range value is defined if we use this section. The range
is:
-1 < y < 1
Remember, to find an inverse, it is the reflection about the y
= x axis.
y = sin-1 x is the notation used to represent the inverse sin function.
It is also referred to as the arcsin. The graph of the inverse function
looks like:
Notice, that the range is now the domain and the domain is now
the range. Because we have restricted the domain, all answers are now
related to the first quadrant or the fourth quadrant. Positive answers
in the first and negative answers in the fourth.
The inverse function of any of the trig functions will return the
angle either measured in degrees or radians. You must be aware that
all positive values will return an angle in the first quadrant and negative
values will return an answer in the fourth quadrant!!
With your calculator set to degree mode:
Sin-1
.81 = 54.1o
Sin-1 (-.2) = -11.5o ( 348.5)
Notice, that domain is: -1 < x < 1.
Taking any other value will result in an error message on your calculator.
The Cos function and it's inverse and the Tan and it's inverse
are also graphed below:
The domain for the inverse cosine is -1 < x <
1, with the range at
0 < y < p.
This means that a positive x value will return an answer in the
first quadrant and a negative x value will return an answer in the second
quadrant.
The domain for the arctan is all real numbers with the range
-p/2 < y < p/2
The arctan will return the values the same way the inverse sine
returns values, in the first and 4th quadrants.
Examples for calculator problems
Find the answers in radian measure. Set calculator mode to
rads.
1) Cos-1 (-.5) = 2.09 rounded to nearest hundredth.
2) Sin-1(-.75) = -.85
3) Tan -1 (5) = 1.38
Find the answers in degree mode. Set calculator to degree
mode.
4) Cos-1 (.8972) = 26.2o
5) Sin-1 (.3333) = 19.5o
6) Tan-1 (3.2) =72.6o
Problems
without using calculator 1) Tan-1 (-1) = x means tan x = -1.
In the fourth quadrant x = -45o or 315o
_
__ 2) Sin-1 ( \/3/2) = x means
that Sin x = \/3/2
In the first quadrant this is 60o 3) Tan(Tan-1 (.5)) = x.
Since .5 is in the domain of the arctan and these function are inverse operations
the answer is .5 4) Cos-1 (Cos 240o) = x
Since 240o is not in the range of arccos, we need to do this
in two steps. Cos 240o = -.5, thus Cos-1 (-.5) = 120o .
Remember, for the inverse cosine, the answer has to come out in the first
or second quadrant!
5)
Cos(Tan-1 (2/3))
Since 2/3 is positive, the tan q = 2/3 with the
angle being in the first quadrant. Thus y = 2 when x =3 which makes
r = \/ 13
___ ___
Thus the cos q = x/r = 3/ \/ 13 =
3 \/ 13 / 13 6) Cos( Sin-1 ( -4/5))
Since the number is negative, the sin q=-4/5 is
in the fourth quadrant. Thus y = -4 and r = 5 which makes x = 3
Thus , the cos q = x/r = 3/5
Notice , we could do the
last two problems without really knowing the size of the angle!!
That's about it for our inverse functions!
Are you ready for the sample test?
Or would you rather head back and study some
more?
Current quizaroo # 7
1) Convert 15p/4 to degrees.
a)
675o
b) 60o
c) 180o
d) 15o
e) 425o
2) Which quadrant or axis
is described for sin > 0 and tan < 0
a) I
b) II
c) III
d) IV
e) x-axis
3) Find the value of sin 3.4 Round
to four decimal places.
a) .0593
b) .9982
c) .5592
d) -.2555
e) -.9668
4) Express in
terms of a reference angle: Tan (-105o)
a) -tan 75o
b) tan 75o
c) cot 75o
d) -cot 75o
e) -tan 105o
5) Find the Cot-1 5.33 rounded to the nearest tenth of a degree.
Notice , we could do the
last two problems without really knowing the size of the angle!!
