In Exercises 19–24,
a. Use the unit circle shown for Exercises 5...

Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. sin 5𝜋/6

Unit circle with coordinates and angles for trigonometric functions in trigonometry course.

Accepted Answer

Hello everybody. I have everything right today today, we're going to be looking at this question that states the circle with a radius of one unit is partitioned into 12 equal arcs where each arc corresponds to specific T values determine the value of the trigonometric function at the given real number T using the X Y coordinates in the diagram or specify that the expression is undefined and also find out the value in part B using even in odd properties of trigonometric functions. Now I'll scroll down a bit so we can look at the entire graph as well as our answer choices. And we see that in the graph, we have a unit circle with a radius of one as a specified with different values assigned according to the trigonometric function of a unit circle. Now all they're asking us really in that question is just use this graph to solve part A which is sign of seven pi divided by six and part B which is sign of negative seven pi divided by six. Now the answers provided are a part A and part B negative squared of two divided by two as choice B says that part A is negative one half and part B is positive one half. As choice C has both part A and B as negative square to three divided by two. And as choice D has part A as positive squared of two divided by two and part B as negative squared of two divided by two. Now we'll tackle this question, part by part. So we'll start with part A and something very important that we have to know is when we're talking about the unit circle, we have a rule that X is equal to cosine data and Y is equal to sin of beta. And this refers to when looking at the different points associated with the unit circle, the Y value will be associated with of that angle of that data. And the X component of the point will be associated with the cosine of the angle. So they told us that we are at seven pi divided by six in part A. Now at seven pi divided seven pi divided by six, we just have to look at our graph. And we see right here that there is a point and that point comes out to be negative square of three divided by two comma negative one half. So our X is negative squared to three divided by two and our Y is negative one half. Therefore sign of seven pi divided by 67 pi divided by six. In this case, is our theta And if we said that sign of theta equals Y, we just have to look at the Y component of seven pi divided by six, which in this case is negative one half. And that would be the answer for part A and then we can move on and look for part A, part B and we'll have a sign of negative seven pi divided by six. And now there's a rule that we know that states that sign of negative theta is equal to negative sign of theta. So using that, we will therefore have that what we had sign sign of negative seven pi divided by six will now have a negative sign of seven pi over divided by six. And since we already have the answer to the sin of seven pi divided by six, this will be equal to negative negative one half. So a negative multiplying a negative will be a positive, which means that we, our finance will be one half. So negative sign of 75 divided by six is equal to one half. And if we just compare that with our answer, we see that answer choice B will match with our answers. So just like that we were able to use the unit circle and its different rules to figure out the sign for part A and the sign for part B. So I hope that was helpful. And until next time

Key Concepts

Unit Circle

Trigonometric Functions

Even and Odd Properties

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