Step by step
The question
Question: solve cos² x − 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 Name it
Look: cos² x and cos x.
Like x² and x in a quadratic.
We call cos x by one letter: u = cos x.
Then the equation is u² − 2u − 3 = 0: a quadratic.
We solve it the usual way.
2 Factor
We need two numbers.
They multiply to −3, and they add to −2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 3, smallest first.
We take the smaller from the bigger.
We need 2.
1 and 3 give 2.
Yes!
With the signs: 1 and −3.
So it is (u + 1)(u − 3) = 0.
3 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In u + 1 = 0, we take away 1 on both sides.
In u − 3 = 0, we add 3 on both sides.
So u = −1 or u = 3.
4 Put cos x back
But u is cos x.
So cos x = −1 or cos x = 3.
Careful!
These are not the answers yet!
They are values of cos x.
We still need the angles.
5 Find every angle
cos x = 3 can never happen: across on the unit circle is always from −1 to 1.
No angle there.
Cos is the across value.
The points with that across value are on an up-and-down line.
The across value is −1 at one point only.
x = 180°.
On the circle: 1 point.
All together, in order: x = 180°.
6 Check
Let's check every answer in cos² x − 2 cos x − 3 = 0.
cos x is −1 at 180°.
Each one must give 0.
Every answer works.
So x = 180°.
Let's look back.
First: name it.
Then: factor.
Then: one bracket is 0.
Then: put cos x back.
Then: find every angle.
Last: check.