Step by step
The question
Question: solve 2 sin² x − 5 sin x + 2 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Name it
Look: sin² x and sin x.
Like x² and x in a quadratic.
We call sin x by one letter: u = sin x.
Then the equation is 2u² − 5u + 2 = 0: a quadratic.
We solve it the usual way.
2 Find two numbers
There is a number in front of u², but it can still factor.
The number in front of u² is 2.
So we multiply 2 by 2: that is 4.
We need two numbers that multiply to 4 and add to −5.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 4, smallest first.
We add each pair.
We need 5.
1 and 4 give 5.
Yes!
With the signs: −1 and −4.
3 Split the middle
We write −5u as −u − 4u.
It is the same amount.
4 Group in pairs
We put the first two together, and the last two together.
From the first pair we take out u.
From the second pair we take out −2.
Both leave the same bracket.
Now both parts have 2u − 1 in them.
5 Same bracket out
2u − 1 is in both parts, so we take it out once.
What is left, u − 2, goes in the other bracket.
So we get (2u − 1)(u − 2).
6 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 2u − 1 = 0, we add 1 on both sides.
In u − 2 = 0, we add 2 on both sides.
Now we have 2u = 1 or u = 2.
To remove times 2, we divide both sides by 2.
So u = 1/2 or u = 2.
7 Put sin x back
But u is sin x.
So sin x = 1/2 or sin x = 2.
Careful!
These are not the answers yet!
They are values of sin x.
We still need the angles.
8 Find every angle
sin x = 2 can never happen: up on the unit circle is always from −1 to 1.
No angle there.
From the unit circle: sin 30° = 1/2.
So the reference angle is 30°.
Sin is the up value.
The points with that height are on a flat line.
Sin is positive above the x-axis: quadrants I and II.
30° and 180° − 30°: x = 30° or 150°.
On the circle: 2 points.
All together, in order: x = 30°, 150°.
9 Check
Let's check every answer in 2 sin² x − 5 sin x + 2 = 0.
sin x is 1/2 at 30°.
Each one must give 0.
Every answer works.
So x = 30°, 150°.
Let's look back.
First: name it.
Then: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Then: put sin x back.
Then: find every angle.
Last: check.