Solve tan^2 x = 3 Step by Step

Step by step

The question

Question: solve tan² x = 3.

Find every angle from 0° up to 360°.

tan² x = 3

We want the angles from 0° to 360°: one full turn.

0° ≤ x < 360°

1 Square root of both sides

tan x squared is 3.

So tan x is the square root of 3, or minus it.

Both work.

tan x = ±√3

Careful!

A square root gives two answers: plus and minus.

tan x = −√3 works too.

Careful: not only tan x = √3 ✗

So tan x = √3 or tan x = −√3.

tan x = √3 or −√3

2 Find the reference angle

From the unit circle: tan 60° = √3.

So the reference angle is 60°.

The minus only changes which quadrants.

tan 60° = √3

3 Find every angle

Tan is positive in quadrants I and III.

60° and 180° + 60°: x = 60° or 240°.

tan x = √3: x = 60° or 240°

Tan is negative in quadrants II and IV.

180° − 60° and 360° − 60°: x = 120° or 300°.

tan x = −√3: x = 120° or 300°

On the circle: 4 points, so 4 angles.

So x = 60°, 120°, 240°, 300°.

x = 60°, 120°, 240°, 300°

4 Check

Let's check every answer in the equation.

Each one must give 3.

x = 60°: tan² 60° = (√3)² = 3

x = 120°: tan² 120° = (−√3)² = 3

x = 240°: tan² 240° = (√3)² = 3

x = 300°: tan² 300° = (−√3)² = 3

Every answer works.

So x = 60°, 120°, 240°, 300°.

Let's look back.

First: square root of both sides.

Then: find the reference angle.

Then: find every angle.

Last: check.

Answer