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:
12 + (√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.