Cosine: Equation

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

Formula

Formula

For cos x = k,
if one of the solution is θ,
then the general solution is
x = 2nπ ± θ.
(n is an integer.)

Example

Example

Solution

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.