In Exercises 44–48, find the reference angle for each angle.

Question

- 11𝜋/3

Accepted Answer

Hello, today we're gonna be calculating the reference angle for are the given angle. So the angle that is given to us is negative 10 pi over three. Now, one thing to note is that this angle is not in standard form. This angle lies outside the one rotation of the unit circle and one rotation of the unit circle is defined from 0 to 2 pi. So we're going to need to rewrite our given angle into standard form or in other words, an angle that lies between zero and two pi in order to find the reference angle, what we can go ahead and do is we can add two pi to negative 10 pi over three to help us rewrite this in an angle that lies between zero and two pi. Now, in this case, two pi over three can be rewritten as six pi over three. So we have negative 10 pi over three plus six pi over three and adding these values together is going to give us negative four pi over three. Now this is still outside one rotation of the unit circle. So we're going to add six pi over three to negative four pi over three, adding these values together is going to give us an angle measure of two pi over three, two pi over three lies between the region of zero and two pi. So in order to get the reference angle for two pi over three, we're going to need to first locate where two pi over three is located at on the unit circle. If you're ever unsure of where radiant is located on the unit circle, you can always multiply the radiant by the quantity 180 over pi doing. This allows us to reduce the pie in the numerator and denominator to just one. And what we are left with is two times 180 which is going to give us the value of 360. And that is going to be over three times one which is three, 360 divided by three is going to give us 120. So what this means is that two pi over three is equivalent to 120 degrees. And the question is where is 120 degrees located on the unit circle? Well, 120 degrees is going to be located in quadrant two. And now that we know the location of 100 and 20 degrees, we can calculate the reference angle. The reference angle lies between our given angle and the X axis. And in order to get a reference angle in quadrant two, we're going to take the angle measure of 180 degrees and subtract 120 degrees from this value 180 minus 120 is going to give us the value of 60 degrees. And this is going to be the reference angle to negative 10 pi over three. But one thing to note is that we want to convert 60 degrees into radiance. In order to do this, we're going to take the value of 60 degrees and multiply it by pi over 1 80 multiplying these values together is going to give us 60 pi over and this fraction is going to reduce to leave us with pi over three. So that means that the reference angle to negative 10 pi over three is going to be pi over three. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 44–48, find the reference angle for each angle.
- 11𝜋/3

Key Concepts

Reference Angle

Angle Reduction Using Coterminal Angles

Quadrants and Sign of Angles

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