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/6)

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

Accepted Answer

Hello everybody. I'm doing all right today. Then we're going to it again 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 and 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 and part B using even and odd properties of trigonometric functions. Now I'm going to scroll down a bit so we can have the entire picture of our graph as well as our answer choices. And we see that they provide us with a unit circle as they mentioned with a radius of one and different points along that circle assigned to them depending on the tri magical function that goes along with it. And we'll see how in a second. Now part A is asking us to solve for the cosine of four pi divided by three. And part B is asking us to solve for the cosine of negative four pi divided by three. Now the choices are s choice A as part A and part B as negative squared of two divided by two. Choice B has part A and part B as negative one half ATO C has part A and part B as positive squared of two divided by two. And as choice D has part A as squared of two divided by two M part V as negative squared of two divided by two. Now we'll tackle this question little bit by little and part by part. So we'll tackle part a first and something that we should note right away when it comes to the unit circle, specifically ingo metric functions is that X is equal to cosine data and Y is equal to sine theta. So basically, when we have an angle data, when we have cosine of the angle, we're talking about the X component. And when we have sign on that angle, we're talking about the Y component and you'll see what I mean in a second. So let's say, for example, they tell us in the question that we are at 54 pi divided by three, that's going to be our theta. Now four pi divided by three, if we look at our graph, we see that there is a point assigned to four pi divided by three and that point is negative one half, comma negative square to three divided by two. So what does this mean? Therefore, we have that cosine of four pi divided by three? Well, four pi divided by three is our data. And we just said that cosine data is equal to X and we have a point associated with this data. So our cosine of four pi divided by three is equal to the X component of that point in this case, negative one half. And that is it for part A. So the unit circle and the relationship with cosine has given us the answer for part A and we can move on to part B where we have cosine of negative four pi divided by three. Now, we can remember another relationship that we know where we have cosine of negative data that is equal to cosine of positive beta. So we can just drop the negative and it, it would be the same. So therefore, cosine of negative four pi divided by three is equal to cosine of positive four pi divided by three. And we already calculated coin of four pi divided by three. So that will be equal to negative one half. And just like that, thanks to the unit circle that they provided us and the relationships that we know we're able to solve for part A and part B and we see that our answers match with answer choice B. So I hope that was helpful. And until next time.

Key Concepts

Unit Circle

Even and Odd Properties of Trigonometric Functions

Trigonometric Function Values

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