In Exercises 25–32, the unit circle has been divided into eigh...

Question

In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

cos 9𝜋/2

Unit circle with coordinates for angles 0, π/2, π, 3π/2, and 2π marked.

cos 9𝜋/2

Accepted Answer

Hello everybody I have doing art today that we're going to be looking at this question that states the circle with a radius of one unit is partitioned into eight equal arcs where each arc corresponds to specific values. Now determine the value of the trigonometric function at the given real number using the X comma Y coordinates in the diagram or specify that the expression is undefined. Also find out the value in part B using periodic properties of trier metric functions. Now I scroll down a bit so we can see our full diagram and our answer choices and we see that we are given a unit circle where we have our, the circle with a radius of one and we have values and uh associated with the trigonometric functions assigned to set circle along that along the coordinate system. Now part A is asking us to solve for a cosine of three pi divided by two. And part B is asking us to solve for a cosine of 11 pi divided by two. Now the answers provided are a part A is negative one half and part B is positive one half as choice B has part A and B, both zero answer choice C has both part A and B as negative square root of three divided by two and choose D has both part A and part B as negative one half. Now let's tackle this part by part. So first we'll focus on part A and something that we should note right away is that when we're talking about the unit circle and the coordinate system associated with trigonometric functions, we should know that X is equal to cosine of data. So when we're talking about cosine, we look at X components and Y is equal to sine of data. So when we're talking about sine, we're talking about the Y component. Now we should also know that pi divided by two is equal to degrees. So let's say you start on the right side, you always start on the right side of the X axis. So to the right of the Y axis, and let's say you move pi divided by two units, you move 90 degrees. So now you are at the top side of the Y axis above the X axis and so on. So if we move, let's say, let's say now we move three pi divided by two, which is what part A is telling us three pi divided by two would be the same thing as moving 270 degrees. And therefore, we will arrive at the bottom half of the Y axis which they tell us has a point at zero comma negative one. So that's the point that we'll be working with. Now, if cosine of data is equal to X, like we said, then cosine of three pi divided by two is equal to zero because our X at three pi divided by two and zero. And just like that, we obtain the answer to part A and we can move on to part B. Now part B will be very similar. And you'll see why in a second. Now we have cosine of pi divided by two. Now this is the same as saying cosine of open parentheses. Four pi plus three pi divided by two. Now, why are these equal to each other? Well, we're dealing with a circle. So pi is 100 and 80 degree turn two pi will be a full 360 degree turn. Now we're at 11 pi. That means that for pie is another turn. Six pie is another turn and so on. However, if we reach four pie and then to get to 11 pie, we do four pie. So that's we go around in a circle four times and then we have to do three pi divided by two to reach our 11 pi divided by two because four pi is the same thing as saying eight pi divided by pi divided by two plus three. Pi divided by two is 11 pi divided by two. So we don't care about the first four pi because you're just going on the circle four times. So all the values repeat, nothing changes. For those four times. You care about where you end up in that last revolution around the circle. In this case, we end up at three pi divided by two. So we have that cosine of pi divided by two is equal to cosine of three pi divided by two. And we just said that co sign up three pi divided by two is equal to zero. So just like that, we obtain the answer for both part A and part B simply by looking at the relationship with the unit circle. Now these answer choice answers match with answer choice B. So that's the answer that we will highlight for this question. So I hope that was helpful. And until next time.

Key Concepts

Unit Circle and Coordinates

Periodic Properties of Trigonometric Functions

Special Angles and Their Coordinates

Watch next