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 47𝜋/4

Accepted Answer

Welcome back everyone. In this problem, we have a circle with a radius of one unit that's partitioned into eight equal arcs where each arc corresponds to specific values. In part a of the question, the sign of three quarters of pi we want to determine the value of the trigonometric function using the X Y coordinates in the diagram or specify that the expression is undefined. In part of the problem, we want to use periodic properties to help us figure out what the sign of 43 4th of pie would be equal to in our answer choices. They're basically a mix of positive or negative values of the root of two divided by two and positive or negative values of a half. Now, what all of this just basically means is that we here have a unit circle that's already split into eight parts and it gives us the X and Y values on the unit circle. Now, how is the unit circle related to the core sign and sign values? Well, recall that for a unit circle, right, then the X value is equal to the core sign of the angle for unit circle. And the Y value is equal to the sign of the angle of the unit circle. And we already have our angles here. We can see 1/4 of pi 3/4 of pi and so on. So if we can take our expression and then figure out which one of the coordinates we should use, then we can get our answers. So let's start with a, the sign of three quarters of pi. First of all, let's find three quarters of pie, which would be right here on at, in between the points negative 10 and 01. Now remember that the sign of the angle is the Y value. So at the angle 3/4 of pi, then the sign would be the root of two divided by two. That means the sign of 3/4 of pi must be equal to the root of two divided by two. Next in choice in, in to be part of the question, we want to find a sign of 43 4th of pie. The problem is we don't have 43 4th of pi on the diagram, but we can recall the periodic property of trigonometric functions which says that after a while trigonometric function values start to repeat. So the period of the unit circle is we can see that right here. So if we can figure out what this angle is, how many times it has it has cycled or it has repeated, then we'll be able to find the value. So first, let's go ahead and divide 43 P by four. No, 43 pi divided by four. If we actually worked the toad, we'd end up with 10 pie because four into 44 divided by four is equal to 10. So that's 10 pi plus a remain of 3/ of pie. No, think about it, right? 10 pie. We want to know how many times did, did we go around? Well, if one period is two pie, that means to get to 10 pie, we'd have to go to two pie and then four pie, six pie, eight pie and then 10 pie. So it means we have five periods. So after that fifth period, after 10 pie, then the values would repeat. Therefore, that tells us that the sign of 43 4th of pie is basically equal to the sign of 3/4 of pie because it's, it has, it's started over and the sign of 3/4 of pi, we already know that that was the square root of two divided by two. Therefore, the sign of 4 to 3/4 of pi is also going to be equal to the square root of two divided by two which the answer choice for that would be a thanks for watching. I hope this helped.

Key Concepts

Unit Circle

Trigonometric Functions

Periodic Properties of Trigonometric Functions

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