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.

cos 3𝜋/4

Video transcript

Welcome back everyone. The circuit with the readers of one unit is partitioned into eight equal arcs where each arc corresponds to specific values determine the value of the trigonometric function at a given real number using the XY coordinates in the diagram or specify that the expression is undefined. We're trying to find the cosecant of seven fourths of pi. And for our answer choices, A says it's negative two B two C root two and D, the negative value of root two no or are we going to use our unit circle to help us to figure out the value of the COSI kind of seven fourths of pi? What do we know about the unit circle? We'll recall that in a unit circle. OK. Then the coordinates of the unit circuit, the XY coordinates can also be written as the cosine of T and the sine of T. In other words, the X coordinate equals CT and the Y coordinate equals ST. Now, how is this supposed to help us? Well, we know that of course, you can't is equal to the reciprocal of the sine function. This case, let's say the Cosic of T equal to the reciprocal of the sine of T. So if we can identify the sine of T, we can reciprocate it to find the cosine of T for our value of T. Now at T equals seven fourths of pi, OK. The value here for the co second, if we go to our unit circle and go to seven fourths of pi and the fourth quadrant notice that its XY coordinates are root two divided by two and the negative root two divided by two. Since the sine function is the Y coordinate, then this tells us that why, which equals sign of seven fourths of pie eh is going to be equal to the negative value of root two divided by two. So what does that mean? Well, this note tells us then that the coi of seventh worth of pie must be equal to the reciprocal of negative root two divided by two which equals negative two divided by root two. Now, if we multiply this expression here, OK. By route two, yeah, if we multiply here, then it's gonna be root two divided by root two. OK. Let me make some space here. That's going to be equal to negative two, root two divided by root two, multiplied by root two, which is simply two. I know two cancers two to give us the co kind of seven fourths of pi to be equal to negative root two. Therefore, D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.