Step by step
The question
Question: solve 2 cos x + √3 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Remove the + √3
To remove + √3, we take away √3.
We do it on both sides, so they stay equal.
+ √3 and − √3 make zero.
Cross them out.
Careful!
The √3 does not just jump across.
We took it away on both sides, so on the right it is − √3.
So 2 cos x = −√3.
A root is a number like any other.
2 Remove the 2
2 cos x means 2 times cos x.
We want just cos x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just cos x is left.
So cos x = −√3/2.
3 Find the reference angle
From the unit circle: cos 30° = √3/2.
So the reference angle is 30°.
The minus only changes which quadrants.
4 Find every angle
Cos is the across value.
The points with the across value we want are on an up-and-down line.
Cos is negative left of the y-axis: quadrants II and III.
180° − 30° and 180° + 30°: x = 150° or 210°.
On the circle: 2 points, so 2 angles.
So x = 150°, 210°.
5 Check
Let's check every answer in the first equation.
Each one must give 0.
Every answer works.
So x = 150°, 210°.
Let's look back.
First: remove the + √3.
Then: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.