# Tangent: Equation

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

## Formula

### Formula

For tan x = k,

if one of the solution is θ,

then the general solution is

x = nπ + θ.

(n is an integer.)

## Example

### Example

### Solution

See tan x = √3 (= √3/1).

Tangent is TOA:

Tangent,

Opposite side (√3),

Adjacent side (1).

So draw a right triangle on a coordinate plane

whose opposite side is √3

and whose adjacent side is 1.

Find the missing side

by using the Pythagorean theorem:

1^{2} + (√3)^{2} = [hypotenuse]^{2}.

Then the hypotenuse is 2.

This is a right triangle

whose sides are 1, √3, and 2.

So this is a 30-60-90 triangle.

So the central angle is, 60º, π/3.

Radian Measure

π/3 is the angle

that starts from 3 o'clock position.

So θ = π/3.

Then the general solution of tan x = √3 is

x = nπ + π/3.

Find the x values

that are in (0 ≤ x ≤ 2π).

n = 0

x = 0⋅π + π/3

= π/3

This is in (0 ≤ x ≤ 2π).

n = 1

x = 1⋅π + π/3

= 4π/3

This is also in (0 ≤ x ≤ 2π).

x = π/3 and x = 4π/3

are in (0 ≤ x ≤ 2π).

So write x = π/3, 4π/3.

So x = π/3, 4π/3 is the answer.