In Exercises 49–59, find the exact value of each expression. D...

Question

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)

Accepted Answer

Hello everybody. I have a doing night today. Today, we are going to look at this question of states determine the exact value of the following expression without the help of the calculator where the expression is sign of 20 pi divided by three. Now the answers provided are a squared of three divided by two B negative squared of three divided by two C negative one half and D one half. Now, the first thing I'm going to do is rewrite our angle because well, I'll show you in one second. So we'll have sign, we have sign of 20 pi divided by three. Now this will be a sign and I'll just rewrite it as a sign of six pie plus two pi divided by three, six pi is 18 pi divided by three. So 18 pi divided by three plus two, pi divided by three is 20 pi divided by three, which is what we started with. Now, why do I write this? Well, remember for trigonometric functions, we're going to be working with a coordinate system working with a unit circle. So if we have 20 pi divided by three and say the left side of the X axis is pi and the right side of the x axis is two pi. So one revolution around that circle would be two pi. Now, if we have six pie, that means we did three revolutions around that circle. And if you're going in a circle, everything repeats. So you don't care about that. Those first three revolutions, you don't care about them because you just went around in a circle three times. What you do care about is that last two pi divided by three because you don't complete that revolution. So that is where that, that is the angle that we're going to be working with. So we will work with sign of two pi divided by three. Now where is two pi divided by three? Well, we have that two pi divided by three is greater than pi divided by two, but less than pie. Now what is pi divided by two pi divided by two is the top half of the y axis. So the 90 degree. So if we're between 90 degrees and high, we are in quadrant two, we are somewhere in quadrant two. Now we wanna look at our reference angle. So we have different rules for the specific quadrants to how to calculate the reference angles in quadrants two. Specifically the reference angle is equal to pi minus the given angle. So we'll have the reference A is equal to pi minus two pi divided by three. So we'll have that the reference angle is equal to pi divided by three. And now we can move on and let's look at our sign and our uh in this case, so when we're working with the unit circle, we should note that the X component is equal to cosine of theta and the Y component is equal to a sign of theta. So it would have a quadrant two. Let's look at the Y component of a graph in quad and two, where are Y values positive? Well, if we look at a coordinate system, Y values are positive above the X axis, so that top half of the Y axis is positive. Now what quadrants lie in that area quadrant one and quadrant two. So we have that sign sign is positive in quadrants two. And that's important to know. So we know the sign of it. So we know if our sign will be negative or positive. Once we get to our answer. Now, using the unit circle, the unit circle tells us that at pi divided by three, we have a point of one half comma squared of three divided by two. So therefore, we have the sign of 20 pie divided by three is equal to sign positive sign as we said of pi divided by three. And as we said, Sine has to do with the Y components. So we just look at the Y value of our points at pi divided by three and that is square root of three divided by two. And that will be the answer for this question which matches with answer choice. A. So as you can see, it's very important to know how to use the unit circle. And how do you really use the rules around it to solve questions like this one? So I hope that was helpful. And until next time.

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)

Key Concepts

Radian Measure and Angle Conversion

Periodic Properties of the Sine Function

Exact Values of Sine for Special Angles

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