Find the exact values of s in the given interval that satisfy ...

Question

Find the exact values of s in the given interval that satisfy the given condition.

[-2π , π) ; 3 tan² s = 1

Accepted Answer

Welcome back. I am so glad you're here. We're told, considering the interval from negative pi to pi inclusive of negative pi but not of pi find the exact values of P. Then we're given that the tangent squared of P equals three. Our answer choices are answer choice. A negative pi divided by six pi divided by six answer choice B negative five pi divided by six negative pi divided by six pi divided by six and five pi divided by six. Answer trace C negative pi divided by four negative three pi divided by four pi divided by four and three pi divided by four and answer trace D negative two pi divided by three, negative pi divided by three pi divided by three and two pi divided by three. All right. So we've got to go a couple different directions here. But first, let's start off getting this so that it's the tangent of P instead of the tangent squared of P, we'll do that by taking the square root of both sides of our equation. And then that's going to give us on the left, the tangent of P that equals on the right because we're taking the square root that's going to be plus or minus the square root of three. OK. Now we're going to switch gears for a moment and we're going to think about the unit circle. We recall from previous lessons that we've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle, going to overlay this with the unit circle where every point on that circle is one unit away from the origin. It is not true of what I drew. But you can imagine that it is now, I am not going to mark every detail of the unit circle. Definitely check out the image of the unit circle in your textbook. But I'm going to label at least the first axis, the quadrant one, excuse me, labeling quadrant one. So closer to the positive part of the X axis than the positive part of the Y axis. The first angle that we mark is pi divided by six which has xy coordinates of square root three divided by 21 half. And then right in the middle between the positive parts of the X and Y axis, we have the angle pi divided by four and those xy coordinates are square root two divided by two square root two divided by two. And then closer to the positive part of the y axis than the positive part of the X axis, we have the angle pi divided by three its xy coordinates are one half square at three divided by two. All right, we'll need that unit circle. We recall from previous lessons that the tangent of an angle will say theta is equal to the Y values divided by the X values. And here our tangent of our angle is equal to plus or minus the square of three. So can we take our Y values? We've got one half square root two divided by two and square root three divided by two and divide those by our, by the corresponding X values and get square root of three? Well, looks like pi divided by four. That angles xy coordinates aren't going to get us a square root three. Our best bet looks like pi divided by three because it has a square root three in the numerator of its Y value. But let's check it. So if we take the square root of three, divided by two, divided by one half, that's the same as square root three divided by two, multiplied by the reciprocal of the second angle which is two, those twos cancel out. So yes, this is the square root of three. So pi divided by three is our angle for P but that's just one of them. So we know P equals pi divided by three. But we've got an interval from negative pi to pi, where is that? Well, if we start from the initial side at the positive part of the X axis and we had clockwise and we're going down through the fourth quadrant, through the third quadrant, negative Pi is going to be the negative part of the X axis. And now we're going from there and now we're going to head counterclockwise as we normally do. And we're going all the way around until we get back to PI, which is the, the negative part of the X axis. So it's one full rotation. But starting in a different spot, we know that pi divided by three is one of our values. It is within that range, but we have to figure out for the other quadrants and pi divided by three is our reference angle for that. So what are, what's which angles is it the reference angle for or in other words, which angles share the same XY coordinates but between negative pi and positive pi. Well, we know that if we're talking about a regular um going starting at zero and going around counterclockwise to two pi in the second quadrant, our angle that uses pi divided by three for a reference angle is closer to the positive part of the y axis than the negative part of the X axis. That one's two pi divided by three and it has those same XY coordinates except negative one half, positive square root three divided by two. And that one's going to get us a negative square root three. But that's one of the things we're looking for, we're looking for plus or minus the square root of three. So two pi divided by three is going to be another one of our responses. If we were to continue counterclockwise in the third quadrant, the angle that uses pi divided by three for its reference angle is closer to the negative part of the y axis than the negative part of the X axis. That is four pi divided by three with those xy coordinates of negative one half negative square root three divided by two. That would get us a positive square root three, which is one of the acceptable answers for the tangent of P. And so that four pi divided by three needs a co terminal angle that is between negative pi and pi. So if we take four pi divided by three and subtract a full rotation to pi, what we're going to get is a negative two pi divided by three. So there is another one of ours. Now in the fourth quadrant, imagine again, we're heading counterclockwise on a full rotation from 0 to 2 pi our angle that uses pi divided by three for a reference angle is going to be closer to the negative part of the Y axis than the positive part of the X axis. That's going to be five pi divided by three with the xy coordinates of one half negative square root three divided by two. That would give us a negative square root three, which is one of the acceptable answers. But because we're between negative pi and positive pi, we need a co terminal angle with five pi divided by three. So we'll subtract two pi from that and we'll get a negative pi divided by three. And those are our four exact values for P when our intervals from negative pi to pi, we look at our answer choices and this matches with answer choice. D well done. We catch you on the next one.

Find the exact values of s in the given interval that satisfy the given condition.

[-2π , π) ; 3 tan² s = 1

Key Concepts

Trigonometric Functions

Solving Trigonometric Equations

Interval Notation

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