# How do you sketch the angle -240 degrees and find its reference angle?

Feb 16, 2018

Add +360 degrees until you have a positive angle, then sketch. The reference angle is the angle from the sketch to the x-axis, in this case, 60 degrees.

#### Explanation:

It makes sense here to state the angle in terms of its positive coterminal angle. To find this, add a positive rotation (360 degrees) until you get a positive angle.

$- 240 + 360 = 120$

Since 120 is positive, you can stop here. A 120 degree angle is in QII, so that's where you'll draw your sketched angle. The reference angle is the angle that gets you to the x-axis. For an angle of 120 degrees, we would subtract that from 180 degrees to find the reference angle:

$180 - 120 = 60$

Your reference angle is therefore 60 degrees.

Feb 16, 2018

The required reference angle is ${180}^{\circ} - {120}^{\circ} = {60}^{\circ}$

#### Explanation:

Given:

color(red)(theta = -240^circ

We need to find the Reference Angle for this given angle.

We are given color(red)(theta = -240^circ

Note that it is a negative angle.

To find the reference angles we can use the chart below:

The chart given above refers to positive angles only.

Coterminal angles are angles in standard position (angles with the initial side on the positive x -axis) that have a common terminal side.

Then, we can find the Coterminal angle and find the reference angle using the chart above.

Please note that the coterminal angle and the given negative angle in our problem always add up to color(blue)(360^circ.

So, our coterminal angle can be calculated as follows:

color(red)(theta = -240^circ

$\Rightarrow - {240}^{\circ} + {360}^{\circ} = {120}^{\circ}$

Hence, our coterminal angle = ${120}^{\circ}$

We will roughly sketch the details as shown below:

Important:

We know that the negative angle color(green)((-240^circ) and the positive coterminal angle color(green)((120^circ) are equal.

Now, we are in a position to find the reference angle using the formula chart.

The required reference angle is ${180}^{\circ} - {120}^{\circ} = {60}^{\circ}$