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

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.
sin 11𝜋/4

Unit circle with coordinates for angles 0, π/4, π/2, and others marked.

sin 11𝜋/4

Accepted Answer

Hello everybody. Hello everybody. I hope you're doing night today today, we're going to look 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 determine the value of the trigonometric function at the given real number using X comma Y coordinates in the diagram or specify that the expression is undefined and also find out the value in part B using periodic properties of tri trigonometric functions. Now scroll down a bit so we can see our entire graph and our answer choices and we see that they give us a unit circle. So basically, it's just as they said, a circle with a unit of one or a radius of unit one that has a trigonometric function values associated with it. So different points along that coordinate system. Now answer choice A says is asking us to solve for sine of five pi divided by four. And answer choice or um part A is asking us to solve for sin of five pi over four divided by four. And part B is asking us to solve for sine of 13 pi divided by four answer choice A has part A as being negative one half and part B as being the square root of two divided by two part or S choice B has part A being negative square root of three divided by two and part B as being the square root of two divided by two. S choice C has part A as the square root of two divided by two and part B as the negative square root of two divided by two. And answer choice D has both part A and B as being the negative square root of two divided by two. Now, something important that we should note here and we'll go part by part. So we'll start with part A first. But something we should know when it comes to the unit circle and the trigonometric functions that go along with it, we should know that X or the X components will be equal to cosine of data. So when we're talking about cosine, we're really referring to the X components of our graph and we also know that Y is equal to sine of data. So when we're talking about Sine, we're talking about the Y components of the unit circle. Now this means that therefore, if what they're asking us for a sign of five pi divided by four, we simply look at the points that they gave us. So they told us right here, five pi divided by four, they gave us its values. So we have that the Y component, since we're talking about sine of five pi over four, they tell us that it is negative square root of two divided by two. And that's all we have to do. For part A, we simply look at the values that they give us in the unit circle now for part B very similar, but we'll do a little bit of calculation first. So we'll look at sign, we have sign of 13 pi divided by four or we won't do calculation, but we'll rewrite this in a better way. So this will be equal to sign of Tu pie plus five pi divided by four. Now, why is it two pi well, a circle like a revolution of this circle is pi every 90 degrees is pi divided by two. So 360 degrees will be pi. Now pi is the same since we're dealing with a denominator of four. Here we have that and and or in this case, our a revolution for our circle will be two pie because every degrees is pi divided by two. So pie is 100 and 80 degrees. Two pie will be 360 degrees. So if we have that two pi equals degrees, we can also say that eight pi divided by four is equal to 360 degrees. Now, if we have eight pi divided by four plus five pi divided by four, we arrive at the 13 pi divided by four that we're trying to solve for. So that's why we can rewrite this as a sign of two pi plus five pi divided by four. And we really only care about that last five pi divided by four because since we're dealing with a circle, all the values repeat as we, we do the, the second revolution of the circle will be, will have the same values as the first revolution in the circle. So we want to ignore that first revolution, which is denoted by the two pi and we only care about the last five pi divided by four since it doesn't complete that full revolution. So we have that sign of 13 pi divided by four is equal to sine of five pi divided by four. And we already calculated this by looking at our graph and we know that it is negative square root of two divided by two. And just like that by looking at our unit circle and doing some transformations and rewriting, we were, we were able to solve both part A and part B and we see that our answer matches with enter choice D. So I hope that was helpful. And until next time.

Key Concepts

Unit Circle

Trigonometric Functions

Periodic Properties

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