Find the reference angle for16𝜋3

Question

Accepted Answer

Hello everybody. I hope we're doing. All right. Today, today we're going to look at this question that states determine the reference angle for the angle given below where the angle given was 19 pi divided by four. Now the answer choices are a pi divided by six B pi divided by two C pi divided by three and D pi divided by four. Now, the first thing I'm going to do is rewrite our angle. So we have 19 pi divided by four. Now remember him with trigonometric functions. We always like to work within a coordinate system using a unit circle. So I'm just gonna draw a circle and accordion system and let's try to visualize this, visualize this a bit. So the top half of the Y axis is pi divided by two. The left half of the X axis is pine. The bottom half of the Y axis is a three pi divided by two and the right side of the X axis is two pi. So one full revolution of our circle would be too high. So if we have 19 pi divided by four, that can be rewritten as four pi plus three pi divided by four. This is, this is the same as saying we did two revolutions of our circle. So two pi four pi would be two circles and that last revolution we can't complete it. So we end at three pi divided by four. Now four pi can be rewritten as 16 pi divided by four. So 16 pi divided by four plus the three pi divided by four is the 19 pi divided by four that we started with. And we care about this because if you have a unit, a circle and you're just going around it in revolutions, all the values will repeat. So that those two first revolutions don't matter in our case because they don't give us any information for our final value. What we care about is that last revolution that we don't complete. So that is the three pi divided by four. So we work with three pi divided by four. Now three pi divided by four. Where does that lie? Well, we know that it is greater than pi divided by two but is less than pie. So we will be somewhere in quads two. So our we will be somewhere in quadrant. So here now in quadrant two, we have a formula for reference angles from angles originating in different quadrants and quadrant two, the reference angle is equal to pi minus the given angle. So we'll have pi minus the given angle was three pi divided by four. That's the one we're working with. So we'll have that the reference angle is equal to pi divided by four. And that would be our final answer which matches with answer choice D. So as you can see, it is very important when it comes to trigonometric functions and the relationships, it's very important to know how the unit circle works. So I hope that was helpful. And until next time.

Find the reference angle for16𝜋3

Key Concepts

Reference Angle

Angle Measurement in Radians

Finding Coterminal Angles

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