Exact value of tan 7π/6 Step by Step

Step by step

The question

Question: find the exact value of tan 7π/6.

1 Change to degrees

The angle is in radians.

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

7π6 = 7 × 180°6

180 divided by 6 is 30.

Times 7 is 210.

180 ÷ 6 = 30

7 × 30 = 210

So 7π/6 = 210°.

7π6 = 210°

2 Find the quadrant

We turn 210° from the positive x-axis, against the clock.

The ray meets the circle at P.

210° is between 180° and 270°.

So it is in quadrant III, the bottom left.

210° is in quadrant III

3 Find the reference angle

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

Here it is 210° − 180°, which is 30°.

210° − 180° = 30°

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 III, P is left of the y-axis: x is negative.

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

quadrant III : x −, y −

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

So tan 210° = tan 30°: the same size as for 30°, and plus.

tan 210° = tan 30°

5 Use the reference angle's value

From the unit circle: tan 30° = √3/3.

tan 30° = √33

So tan 7π/6 = √3/3.

tan 7π6 = √33

6 Check

Let's check with the calculator: tan 210° is about 0.5774.

tan 210° ≈ 0.5774

And √3/3 is about 0.5774.

The same!

So it is right.

√33 ≈ 0.5774

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