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
angle either measured in degrees or radians. You must be aware
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:
.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
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
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
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
4) Cos-1 (.8972) = 26.2o
5) Sin-1 (.3333) = 19.5o
6) Tan-1 (3.2) =72.6o
without using calculator 1) Tan-1 (-1) = x means tan x =
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!
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
Current quizaroo # 7
1) Convert 15p/4 to degrees.
2) Which quadrant or
axis is described for sin > 0 and tan < 0
3) Find the value of sin 3.4
Round to four decimal places.
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.