Solving Arctangent Functions

Solving Arctangent Functions

How to solve arctangent function problems: definition, example, and its solution.

Definition

y = arctan x is the inverse function of y = tan x (-pi/2 < x < pi/2).

y = arctan x is the inverse function of y = tan x.

Of course the graph of y = tan x
is not one-to-one:
it fails the horizontal line test.

So, to define the inverse function,
the domain of y = tan x is fixed like this:
-π/2 < x < π/2.

Graphing tangent functions

Example

Find the value of the given expression. cos (arctan [-sqrt(3)])

Set x = arctan (-√3).

Next, tangent both sides.
(left side) = tan x
(right side) = tan (arctan [-√3])
= -√3

So tan x = -√3.
(-π/2 < x < π/2)

To solve tan x = -√3,
draw a right triangle that satisfies
tan x = -√3 / 1.

Tangent: TOA.
So tan x = (opposite side) / (adjacent side) = -√3 / 1.

The right triangle is a 30-60-90 triangle.

So (hypotenuse) = 2.

So arctan (-√3) = x
= -π/3.

So cos (arctan (-√3)) = cos x
= cos (-π/3).

Then, from the right triangle,
cos (-π/3) = 1/2.

Actually, you don't need to write x = -π/3
for finding cos x,
because from the right triangle,
you can directly find cos x = 1/2.

Just wanted to show you
arctan (-√3) = -π/3.