Solve 2 cos^2 x - 3 cos x = -1 on 0 to 2π Step by Step

Step by step

The question

Question: solve 2 cos² x − 3 cos x = −1.

Find every angle from 0 up to 2π.

2 cos² x − 3 cos x = −1

From 0 to 2π: one full turn.

We work in degrees, and change to radians at the end.

0 ≤ x < 2π

1 Get 0 on one side

Everything goes on the left, so the right side is 0.

We take away −1 on both sides, so they stay equal.

2 cos² x − 3 cos x + 1 = 0

2 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.

u = cos x

Then the equation is 2u² − 3u + 1 = 0: a quadratic.

We solve it the usual way.

2u² − 3u + 1 = 0

3 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 1: that is 2.

We need two numbers that multiply to 2 and add to −3.

2 × 1 = 2

? × ? = 2, ? + ? = −3

They multiply to a plus number, so the two numbers have the same sign.

They add to a minus number, so both are minus.

× is + → same signs: both −

We list the pairs that make 2, smallest first.

We add each pair.

We need 3.

1 and 2 give 3.

Yes!

With the signs: −1 and −2.

1 × 2: 1 + 2 = 3 ✓

4 Split the middle

We write −3u as −u − 2u.

It is the same amount.

2u² − u − 2u + 1 = 0

5 Group in pairs

We put the first two together, and the last two together.

(2u² − u) + (−2u + 1) = 0

From the first pair we take out u.

From the second pair we take out −1.

Both leave the same bracket.

(2u² − u) = u(2u − 1)

(−2u + 1) = −1(2u − 1)

Now both parts have 2u − 1 in them.

u(2u − 1) − 1(2u − 1) = 0

6 Same bracket out

2u − 1 is in both parts, so we take it out once.

What is left, u − 1, goes in the other bracket.

ab + cb = b(a + c)

So we get (2u − 1)(u − 1).

(2u − 1)(u − 1) = 0

7 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

2u − 1 = 0 or u − 1 = 0

In 2u − 1 = 0, we add 1 on both sides.

In u − 1 = 0, we add 1 on both sides.

2u − 1 + 1 = 0 + 1 or u − 1 + 1 = 0 + 1

Now we have 2u = 1 or u = 1.

2u = 1 or u = 1

To remove times 2, we divide both sides by 2.

So u = 1/2 or u = 1.

u = 12 or u = 1

8 Put cos x back

But u is cos x.

So cos x = 1/2 or cos x = 1.

cos x = 12 or 1

Careful!

These are not the answers yet!

They are values of cos x.

We still need the angles.

Careful: not x = 12 or 1 ✗

9 Find every angle

From the unit circle: cos 60° = 1/2.

So the reference angle is 60°.

cos 60° = 12

Cos is the across value.

The points with that across value are on an up-and-down line.

Cos is positive right of the y-axis: quadrants I and IV.

60° and 360° − 60°: x = 60° or 300°.

cos x = 12: x = 60° or 300°

The across value is 1 at one point only.

x = 0°.

cos x = 1: x = 0°

On the circle: 3 points.

10 In radians

In radians: each angle times π over 180.

0° is 0.

60° is π/3.

300° is 5π/3.

x = 0, π3, 5π3

11 Check

Let's check every answer in 2 cos² x − 3 cos x + 1 = 0.

cos x is 1 at 0°.

Each one must give 0.

x = 0°: 2(1)² − 3(1) + 1 = 2 − 3 + 1 = 0

x = 60°: 2(12)² − 3(12) + 1 = 12 − 32 + 1 = 0

x = 300°: 2(12)² − 3(12) + 1 = 12 − 32 + 1 = 0

Every answer works.

So x = 0, π/3, 5π/3.

Let's look back.

First: get 0 on one side.

Then: 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 cos x back.

Then: find every angle.

Then: in radians.

Last: check.

Answer