Solving Arctangent Functions
How to solve arctangent function problems: definition, example, and its solution.
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
Set x = arctan (-√3).
Next, tangent both sides.
(left side) = tan x
(right side) = tan (arctan [-√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.
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
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.