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. cos 𝜋/3

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

Accepted Answer

Hello everybody I have tonight today. Today we're going to look 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. Also find out the value in part B using even and odd properties of trigonometric functions. Now I'm going to scroll down a bit. So we have a full picture of our graph and our answers and we can see that they provide us with a unit circle with a radius of one as they mentioned, where there are different points along that circle corresponding to different trigonometric functions. As we'll see. Now, in part A, they're asking us to solve for the cosine of seven pi divided by six. And in part B they're asking us to solve for the cosine of negative seven pi divided by six. Answer your choice. A says that part A and part B are both negative squared of two divided by two. As choice B says that part A and part B are both negative one half. As the choice C says that both part A and part B are negative squared of three divided by two. And as choice D says that both, that part A is positive squared of two divided by two and part B is negative squared of two divided by two. Now we will be tackling this question by part part by part. So first we'll focus on part A and when it comes to the unit circle, which is what we have here, there's something very important that we should know. And that is that X is equal to cosine data and Y is equal to sign of the. Now, this just means that when we're talking about cosine of a certain angle in that unit circle, it will co the X that cosine will correspond to the X value. And when we're talking about the sign of that angle, the sign will correspond to the Y value. For example, they tell us that for part A, we are at seven pi divided by six. And the point, if we look at our, our graph at seven pi divided by six, we see that they give us a point that goes with it. And that point is negative square to three divided by two comma negative one half. So how does that point help us? Well, we will therefore have that co sign of seven pi divided by six. We said we said that cosine of theta is equal to X, our, our theta here is 75 divided by six. So we just look at the X, at that point in this case that X is negative squared of three divided by two. And that's it, that's it. For part A, we just look at the unit circle and the value they gave us and that's all we have to do. So now we move on to part B where we have cosine of negative seven pi divided by six. And we should know that there is a relationship where we have cosine of negative theta that will be equal to cosine of positive data. So what does this mean? We will therefore have that cosine of negative seven pi divided by six? That's the same thing as saying cosine of seven pi positive seven pi divided by six. So we are already solved for the cosine of seven pi divided by six. So we have that this will be equal to negative square to three. They got it right. So, and that is the relationship that we have to a cosine of negative data and positive data and just like that with that relationship. And thanks to the unit circle that they provided us, we have been able to solve for both part A and part B. And we see that our answer choice will be answer choice C. So I hope that was helpful. And until next time

Key Concepts

Unit Circle

Trigonometric Functions

Even and Odd Properties

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