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 one-to-one.

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).

Using a Calculator

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.