Exact value of cos 2π/3 Step by Step

Step by step

The question

Question: find the exact value of cos 2π/3.

1 Change to degrees

The angle is in radians.

π is 180°, so we put 180° in place of π.

2π3 = 2 × 180°3

180 divided by 3 is 60.

Times 2 is 120.

180 ÷ 3 = 60

2 × 60 = 120

So 2π/3 = 120°.

2π3 = 120°

2 Find the quadrant

We turn 120° from the positive x-axis, against the clock.

The ray meets the circle at P.

120° is between 90° and 180°.

So it is in quadrant II, the top left.

120° is in quadrant II

3 Find the reference angle

The reference angle is the angle between the ray and the x-axis.

Here it is 180° − 120°, which is 60°.

180° − 120° = 60°

Careful!

Always measure to the x-axis, never to the y-axis.

The y-axis gives 30°: the wrong triangle.

Careful: not 120° − 90° = 30° ✗

4 Find the sign

In quadrant II, P is left of the y-axis: x is negative.

P is above the x-axis: y is positive.

quadrant II : x −, y +

Cos is the across number x.

In quadrant II, x is negative.

So cos 120° = −cos 60°: the same size as for 60°, with a minus.

cos 120° = −cos 60°

5 Use the reference angle's value

From the unit circle: cos 60° = 1/2.

cos 60° = 12

So cos 2π/3 = −1/2.

cos 2π3 = −12

6 Check

Let's check with the calculator: cos 120° is about −0.5000.

cos 120° ≈ −0.5000

And −1/2 is about −0.5000.

The same!

So it is right.

−12 ≈ −0.5000

Let's look back.

First: change to degrees.

Then: find the quadrant.

Then: find the reference angle.

Then: find the sign.

Then: use the reference angle's value.

Last: check.

Answer