In Exercises 35–60, find the reference angle for each angle.

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Question

In Exercises 35–60, find the reference angle for each angle.

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Accepted Answer

Hello, today we're gonna be calculating the reference angle for negative 53 point pi over 12. Now, one thing to note is that negative 53 pi over 12 lies outside the region of one rotation of the unit circle. We're going to have to write this angle in standard form so that it is an angle that lies between zero and two pi. In order to do this, we're going to be having to add two pi to this angle until it reaches a measurement that lies between zero and two pi. So we can rewrite two pi as 24 pi over and negative 53 pi over 12 plus 24 pi over 12 is equal to negative 29 pi over 12. This is still not in standard form. So we're going to keep adding increments of 24 pi over 12 until it reaches standard form. So negative 29 pi over 12 plus 24 pi over 12 is going to give us the value of negative five pi over 12. And if we add one more increment of 24 pi over 12, we get negative five pi over 12 plus 24 pi over 12 is equal to 19 pi over 12. Now, this value has to lie within zero and two pi. And one way that we can check the degree of 19 pi over 12 is to convert it from radiance to degrees. In order to do this, we take 19 pi over 12 and multiply it by the quantity 1 80 over pi doing so allows us to reduce pie to one. And what we are left with is 19 times 180 which gives us 3420 over 12 times one which is 12, 3420 divided by 12 is going to give us 285 degrees. So this is the standard form of negative 53 pi over 12. The next thing we wanna do is we want to go ahead and identify where degrees lies on the unit circle. So if we draw a unit circle, we need to ask ourselves where is 285 degrees located at? While 285 degrees is located in the fourth quadrant of the unit circle. And now what we wanna do is you want to calculate the reference angle, the reference angle always lies between the given angle and the x axis. And in order to get the reference angle for an angle that lies in quadrant four, we're going to have to take 360 degrees and subtract the given angle which in this case is 285 degrees. So 360 degrees minus 285 degrees is going to give us 75 degrees. So that means that the reference angle is 75 degrees for negative 53 pi over 12. Now, what we're going to do is we're gonna convert 75 degrees back into radiance in order to do this, we're going to take 75 degrees and multiply it by the quantity pi over 1 80 75 times pi over 1 80 is going to give us 75 pi over 180 and this quantity is going to simplify to give us five pie over 12. So five pi over 12 is going to be the reference angle for negative 53 pi over 12. 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 35–60, find the reference angle for each angle. _ 13𝜋 3

Key Concepts

Reference Angle

Angle Measurement in Radians

Finding Coterminal Angles

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