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

Step by step

The question

Question: solve 2 cos² x + cos x − 1 = 0.

Find every angle from 0 up to 2π.

2 cos² x + cos x − 1 = 0

From 0 to 2π: one full turn.

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

0 ≤ x < 2π

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.

u = cos x

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

We solve it the usual way.

2u² + u − 1 = 0

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 −1: that is −2.

We need two numbers that multiply to −2 and add to 1.

2 × (−1) = −2

? × ? = −2, ? + ? = 1

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

× is − → one +, one −; bigger one +

We list the pairs that make 2, smallest first.

We take the smaller from the bigger.

We need 1.

1 and 2 give 1.

Yes!

With the signs: −1 and 2.

1 × 2: 2 − 1 = 1 ✓

3 Split the middle

We write u as −u + 2u.

It is the same amount.

2u² − u + 2u − 1 = 0

4 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

5 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

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.

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 take away 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

7 Put cos x back

But u is cos x.

So cos x = −1 or cos x = 1/2.

cos x = −1 or 12

Careful!

These are not the answers yet!

They are values of cos x.

We still need the angles.

Careful: not x = −1 or 12 ✗

8 Find every angle

From the unit circle: cos 60° = 1/2.

So the reference angle is 60°.

The minus only changes which quadrants.

cos 60° = 12

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°.

cos x = −1: x = 180°

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°

On the circle: 3 points.

9 In radians

In radians: each angle times π over 180.

60° is π/3.

180° is π.

300° is 5π/3.

x = π3, π, 5π3

10 Check

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

cos x is 1/2 at 60°.

Each one must give 0.

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

x = 180°: 2(−1)² + (−1) − 1 = 2 − 1 − 1 = 0

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

Every answer works.

So x = π/3, π, 5π/3.

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 cos x back.

Then: find every angle.

Then: in radians.

Last: check.

Answer