Find the exact value of s in the given interval that has the g...

Question

Find the exact value of s in the given interval that has the given circular function value.

[ π , 3π/2] ; sec s = ―2√3/3

Accepted Answer

Welcome back. I am so glad you're here. We're told, considering the interval from pi to three pi divided by two, find the exact value of X. And we're given that the C can't of X equals negative square at two. Our answer choices are answer choice. A pi divided by four answer choice B three pi divided by four answer choice C five pi divided by four and answer choice D seven pi divided by four. All right. So a few things to note here, we're looking for an angle measure and we're told it's within the interval from pi to three pi divided by two. Now, I'm going to draw a quick sketch of a unit circle. Definitely check out the unit circle in your textbook for one with all the details. I'm not going to include all of them. We have a vertical y axis and a horizontal X axis. They come together at the origin in the middle. They are overlain with a unit circle. The unit circle has every point on its circle, one unit away from the origin, which is not true of what I drew. But you can imagine that it is now pi is the angle on the unit circle. That is the negative part of the x axis and three pi divided by two is the angle on the unit circle that is the negative part of the y axis. So for that interval, we're talking about quadrant three down into the left of the origin. Now I'm going to label the angles that are in quadrant three on the unit circle. So we have closer to the negative part of the X axis than the negative part of the Y axis seven pi divided by six. And its xy coordinates are negative square at three divided by 21 half. Then directly in the middle of the third quadrant, we have the angle five pi divided by four and its XY coordinates are negative square root two divided by two negative square root two divided by two. And then I'm sorry for seven pi divided by six, the Y coordinate is negative one half I wrote positive but it's negative. And then the third angle on the unit circle in quadrant three, closer to the negative part of the Y axis than the negative part of the X axis is four pi divided by three. Its xy coordinates are negative one half, negative square root three divided by two. And so that's one thing. Now we recall from previous lessons that the sea can't of an angle on the unit circle. We'll call that angle S is equal to one divided by the X value. So here we're concerned about the X values. So for seven pi divided by six, that's negative square root three, divided by two, for five pi divided by four, that's negative square root two, divided by two. And for a four pi divided by three, that's negative one half. And we're told that one divided by the X value is going to equal negative square root two. That's our exact value. Well, the closest thing we have to a negative square root two is going to be the X value for five pi divided by four. So let's see if we put that in the denominator for our C can. So we have one divided by the X value negative square root two divided by two. Let's see if that simplifies to be negative square root two. So this is the same as one divided by, with the division symbol negative square root two divided by two and dividing with fractions is the same as multiplying by the reciprocal. So this is one multiplied by the reciprocal is going to be two divided by square root two. And remembering that that's negative. So multiplying straight across one multiplied by negative two is a negative two divided by square root two. But we need to rationalize this. So we'll multiply both the numerator and the denominator by square it two in the numerator that's going to be negative two square root two divided by, in the denominator square root two multiplied by itself is just two. Now those twos do cancel out. And so this does simplify to be negative square root two. So our angle that we're talking about here is the five pi divided by four because it's X value when put in the denominator for one, divided by X is equal to negative square root two. So our angle that we found and the angle here is X is equal to five pi divided by four. We look at our answer choices and that matches with answer choice. C well done. We'll catch you on the next one.

Find the exact value of s in the given interval that has the given circular function value.
[ π , 3π/2] ; sec s = ―2√3/3

Key Concepts

Secant Function

Unit Circle

Quadrants and Angle Ranges

Watch next