# Cosine: Equation

How to solve the cosine equation cos x = k: general solution formula, 1 example, and its solution.

## Formula

For cos x = k,

if one of the solution is θ,

then the general solution is

x = 2nπ ± θ.

(n is an integer.)

## Examplecos x = -√2/2

First change -√2/2 to -1/√2,

in order to make the sides of the right triangle simpler.

See cos x = -1/√2.

Cosine is CAH:

Cosine,

Adjacent side (-1),

Hypotenuse (√2).

So draw a right triangle on a coordinate plane

whose adjacent side is -1

and whose hypotenuse is √2.

Find the missing side

by using the Pythagorean theorem:

[height]^{2} + (-1)^{2} = (√2)^{2}.

Then the height is 1.

This is a right triangle

whose sides are 1, -1, and √2.

So this is a 45-45-90 triangle.

So the central angle is, 45º, π/4.

Radian Measure

π/4 is the angle

that does not start from 3 o'clock position.

Then the angle that starts from 3 o'clock is

π - π/4 = 3π/4.

So θ = 3π/4.

θ = 3π/4

Then the general solution of cos x = -1/√2 is

x = 2nπ ± 3π/4.

Find the x values

that are in (0 ≤ x ≤ 2π).

n = 0

x = 2⋅0⋅π ± 3π/4

= ± 3π/4

x should be in (0 ≤ x ≤ 2π).

So x = 3π/4.

n = 1

x = 2⋅1⋅π ± 3π/4

= 8π/4 ± 3π/4

8π/4 + 3π/4 = 11π/4

8π/4 - 3π/4 = 5π/4

x should be in (0 ≤ x ≤ 2π).

So x = 5π/4.

x = 3π/4 and x = 5π/4

are in (0 ≤ x ≤ 2π).

So write x = 3π/4, 5π/4.

So

x = 3π/4, 5π/4

is the answer.