Exact value of tan -45° Step by Step

Step by step

The question

Question: find the exact value of tan −45°.

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: 315°.

−45° + 360° = 315°

Same place on the circle, same point.

So tan −45° = tan 315°.

tan −45° = tan 315°

2 Find the quadrant

We turn 315° from the positive x-axis, against the clock.

The ray meets the circle at P.

315° is between 270° and 360°.

So it is in quadrant IV, the bottom right.

315° 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° − 315°, which is 45°.

360° − 315° = 45°

Careful!

The reference angle is always measured to the x-axis, never to the y-axis.

Careful: measure to the x-axis

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 −

Tan is y divided by x: minus over plus is minus.

So tan 315° = −tan 45°: the same size as for 45°, with a minus.

tan 315° = −tan 45°

5 Use the reference angle's value

From the unit circle: tan 45° = 1.

tan 45° = 1

So tan −45° = −1.

tan −45° = −1

6 Check

Let's check with the calculator: tan −45° is about −1.0000.

tan −45° ≈ −1.0000

And −1 is about −1.0000.

The same!

So it is right.

−1 ≈ −1.0000

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