Inverse Trigonometric Functions

With many of the previous elementary functions, we able to create inverse functions. For example:
the inverse of a linear function is another linear function;
the inverse of the quadratic function is (with restricted domain) the square root function;
the inverse of an exponential function is a logarithm.
And so on....
4.6.1 The inverse of a trig function
Here we examine inverse functions for the six basic trig functions.
Recall that if we are going to take a function f(x) and create the inverse function f??1(x) then the
function f(x) needs to be one-to-one. We cannot have two dierent inputs a and b where y = f(a) = f(b)
for then we don t know how to compute f??1(y): Visually, this says that the graph of y = f(x) must pass
the horizontal line test.
This is a signicant problem for the trig functions since all the trig functions are periodic and so,
given any y-value, there are an innite number of x-values such that y = f(x): Trig functions badly fail
the horizontal line test! We will x this problem (in the next section) by appropriately restricting the
domain of the trig functions in order to create inverse functions.
Before we go deeper into describing inverse trig functions, let us take a moment to review the inverse
function concept. In the past we used the superscript ??1 to indicate an inverse function, writing f??1(x)
to mean the inverse function of f(x): We will continue to do this, writing sin??1 x for the inverse sine
function and tan??1 x for the inverse function of tangent. Etc. But there is another common notation for
inverse functions in trigonometry. It is common to write \arc " to indicate an inverse function, since the
output of an inverse function is the angle (arc) which goes with the trig value. For example, the inverse
function of sin(x) is written either sin??1(x) or arcsin(x). In these notes these terms are equivalent.
Before we go into detail on the inverse trig functions, let s practice the concept of an inverse function.
A Worked Problem
1. Find (without a calculator) the exact values of the following:
(a) arccos(
p
2
2 )
(b) arccos(??
p
3
2 )
(c) arcsin(
p
3
2 )
(d) arctan(1)
(e) arctan(
p
3)
(f) arctan(??
p
3)
Solutions.
(a) Since arccos(x) is the inverse function of cos(x) then we seek here an angle  whose cosine is p
2
2 : Since cos( 
4 ) =
p
2
2 then arccos(
p
2
2 ) should be 
4 :
(b) arccos(??
p
3
2 ) = 5
6 since cos( 5
6 ) = ??
p
3
2 :
(c) arcsin(
p
3
2 ) = 
3 since sin( 
3 ) =
p
3
2 :
(d) arctan(1) = 
4 since tan( 
4 ) = 1:
(e) arctan(
p
3) = 
3 since tan( 
3 ) =
p
3:
(f) arctan(??
p
3) = ??
3 since tan(??
3 ) = ??
p
3:
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