Exact value of cos -60° Step by Step

Step by step

The question

Question: find the exact value of cos −60°.

1 Bring it into 0° to 360°

A negative angle turns the other way, with the clock.

Adding a full turn, 360°, lands at the same place: 300°.

−60° + 360° = 300°

Same place on the circle, same point.

So cos −60° = cos 300°.

cos −60° = cos 300°

2 Find the quadrant

We turn 300° from the positive x-axis, against the clock.

The ray meets the circle at P.

300° is between 270° and 360°.

So it is in quadrant IV, the bottom right.

300° is in quadrant IV

3 Find the reference angle

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

Here it is 360° − 300°, which is 60°.

360° − 300° = 60°

Careful!

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

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

Careful: not 300° − 270° = 30° ✗

4 Find the sign

In quadrant IV, P is right of the y-axis: x is positive.

P is below the x-axis: y is negative.

quadrant IV : x +, y −

Cos is the across number x.

In quadrant IV, x is positive.

So cos 300° = cos 60°: the same size as for 60°, and plus.

cos 300° = cos 60°

5 Use the reference angle's value

From the unit circle: cos 60° = 1/2.

cos 60° = 12

So cos −60° = 1/2.

cos −60° = 12

6 Check

Let's check with the calculator: cos −60° is about 0.5000.

cos −60° ≈ 0.5000

And 1/2 is about 0.5000.

The same!

So it is right.

12 ≈ 0.5000

Let's look back.

First: bring it into 0° to 360°.

Then: find the quadrant.

Then: find the reference angle.

Then: find the sign.

Then: use the reference angle's value.

Last: check.

Answer