6.2 Inverse Tangent and Cotangent
We can now apply the same methods used for inverse sine and cosine
to construct inverses for tangent and cotangent. As before, the
important step is to limit the domains so that the trigonometric
functions become onetoone.
Inverse Tangent and Cotangent



Defining tan^{ 1}(x) and cot^{ 1}(x)
Using the procedures above we arrive at definitions for these two
inverse trigonometric functions.
For any x, tan^{ 1}(x)
is the angle measure in the interval
(/2 , /2)
whose tangent value is x.

For any x, cot^{ 1}(x)
is the angle measure in the interval
(0 , ) whose cotangent value is x.

Inverse Properties
We have the usual composition formulas.
tan^{ 1}(tan(x)) = x for x in the interval
(/2 , /2).
tan(tan^{ 1}(x)) = x for any x.

cot^{ 1}(cot(x)) = x for x in the interval
(0 , ).
cot(cot^{ 1}(x)) = x for any x.

Because of the intervals chosen we get this identity, similar to the one stated
for inverse sine and cosine.
cot^{ 1}(x) = /2 
tan^{ 1}(x)
for any x.

To get used to thinking inversely, try this exercise without a calculator.
The answers involve familiar angles and can be found just using a
sketch of the angles or graphs.
Inverses and Familiar Angles



Using a calculator
It is also important to be proficient at using your calculator to find
inverse values. While calculators can compute the inverse sine, cosine,
and tangent, they usually do not have buttons for the inverses of cotangent,
cosecant, and secant. To compute these inverse function values, you have to
use their relationships to the other three functions.
Example:

Find cot^{1}(2.5) rounded to four decimal places.

Solution: 
One approach is to use the identity
cot^{1}(x) = /2 
tan^{1}(x). Using a calculator set in radian
mode, we find that
tan^{1}(2.5) = 1.19028995. Using
3.14159 we get
cot^{1}(2.5) = (3.14159)/2 − 1.19028995 = 0.3805 (to four decimal places).

We are going to rework this example using a different approach.
Example:

Find cot^{1}(2.5) rounded to four decimal places.

Solution: 
Let y = cot^{1}(2.5) so that cot(y) = 2.5.
Then, tan(y) = 1/2.5 = 0.4. We need to solve this equation. Remember
that there are two solutions  a first quadrant angle and a third quadrant
angle. By calculator, y = 0.3805 or 3.5221. Since the inverse
cotangent must be
a first or second quadrant angle, we need to take the solution of 0.3805.

Use radian mode on your calculator and round answers to four decimal
places in the next exercise. Since most calculators don't have an
"cot^{ 1}" button, you will have to use the fact that if
cot(A) = x then tan(A) = 1/x or the fact that
cot^{ 1}(x) = /2 
tan^{ 1}(x).
Other notation for tan^{1} include arctan and atan. Similarly,
arccot and acot are used for cot^{1}.
Typically, we use atan and acot in the
interactive exercises and demonstrations because of ease of typing and for
computer recognition of the function.