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.

tan 𝜋

Unit circle with coordinates for angles 0, π/4, π/2, π, 3π/4, 2π, 5π/4, 3π/2, and 7π/4.

tan 𝜋

Accepted Answer

Tell everybody I'm reading right today that we're going to look at this question that states the circle with the radius of one unit is partitioned into eight equal arcs where each part corresponds to specific values and determine the value of the trigonometric function at the given real number using the X comma Y coordinate in the diagram or specify that the expression is undefined. Also find out the value in part B using periodic properties of trigonometric functions. Now, if I scroll down a bit, we can see our entire answer and we see that they give us a unit circle. So that's a circle around the X and Y coordinate system labeled with different points along throughout that system. Now, we also see that for part A, they're asking us to solve for sea cans of pie and for part B they're asking us to solve for sea cans of 13 pi. Now the answers provided are ac camp pi or part A is negative one and part B is negative one choice B says that part A is zero and part B is negative one as a choice C has part A and part B both being zero. And as sure as D has part A being negative one and part B being zero. No, when we're trying to solve this question, there's two, a couple of things we should know right away. So we're looking at a circle like this and how we relate it to trigonometric functions. We should know that the X component is equal to cosine of data and that the Y component is equal to sign of the. Now this is using the unit circle method because what we are provided is a unit circle. It's a radi a un a circle with a radius of one related to the different values using trigonometric functions. Now, we should also know that pi divided by two is the same as saying degrees. And why is this important? Well, because they're asking for a second of pi. So if pi divided by two is 90 degrees, pi is equal to 180 degrees. Therefore, if we look at our circle and we start on the right side of the X axis and we move 180 degrees, we arrive at the left side of left hand side of the X axis, which right here they give us a point that states it is negative one comma zero. Now we have that if cosine of theta is equal to X, then C can of theta is equal to one divided by cosine of theta which is equal to one divided by X. Therefore, C cans of pi is equal to one divided by our X component of pi which is negative one, which is equal to one divided by negative one, which is negative one. And just like that, we've obtained our answer for part ac cans of pi is equal to negative one. Now part B will be very similar to this. Although it has a different value, you'll see why it's very similar in a second. So we have sea cans of a 13 pie. Now we have to think about what we've said about pie divided by two and pie and such. So pi divided by two is 90 degrees. If pie is 18, 180 degrees, then two pie is 360 degrees. And we're right back to where we started. And the circle will repeat if we do four pi let's say now if we have 13 pie, that means that's the same thing as saying C cans of pie plus pie. Now, why 12 pie? Well, because 12 pie means we went around the circle three times or uh six times. So we did a full circle six times and now we're missing the that last pie. So what we care about is where we end up because if we go around the circle six times those values repeat, the circle continues to be the same. But if we on the last revolution, we only do pie So when we have a circle, we care about where we end up there. So we have that sea cans of 13 pie is equal to C cans of pie. Because as we just said, the 1st 12 pie of that is just revolving around the circle. And we just said that C can of pi is negative one and just like that, we have the answer for part B as well. Now we have that both answers are negative one which matches with answer, answer choice. A and just like that, using the relationship with the unit circle, we were able to solve both parts. So I hope that was helpful. And until next time.

Key Concepts

Unit Circle and Coordinates

Tangent Function on the Unit Circle

Periodicity of Trigonometric Functions

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