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/2, π] ; sin s = 1/2

Accepted Answer

Welcome back. I am so glad you're here. We're asked to find the exact value of P. The angle P lies in the interval from pi divided by two to pi. We're told that the sine of P equals the square root of three divided by two. Our answer choices are answer choice. A pi divided by three answer choice B two pi divided by three answer choice C five pi divided by six and answer choice D four pi divided by three. All right. So a few things here, we're trying to find an angle measure and we know that the sine of this angle is equal to this exact value square at three divided by two. And that's pretty familiar. So this might very well be on the unit circle. Let's draw a quick sketch of the unit circle. This one is not going to include all of the details. So definitely check out the unit circle figure in your textbook, we have a vertical y axis, the horizontal x axis, they come together the origin in the middle overlying that is the unit circle. An actual unit circle has every point on the circle. One unit away from the origin. You can imagine that mine has that. So we're told that we're in the interval from pi divided by two to pi, well, pi divided by two, that's the positive part of the Y axis and pi that's the negative part of the X axis. So we are in the second quadrant here. So I'm only going to draw the parts of the unit circle that are in the second quadrant. So up in the second quadrant, closer to the positive part of the Y axis than the negative part of the X axis, we have the angle two pi divided by three which is 120 degrees. And then directly in between the positive part of the Y axis and the negative part of the X axis, we have the angle three pi divided by four, that's 135 degrees. And then closer to the negative part of the X axis than on the positive part of the Y axis. We have the angle five pi divided by six, which is 150 degrees. And we recall from previous lessons that the sign of any angle on the unit circle, we can call that s is equal to the Y value. So our X and Y values, let's start with five pi divided by six, that is negative square root three, divided by two for the X value and one half for the Y value. So the Y value there is one half, not square root three divided by two. For three pi divided by four, the X and Y values are negative square root two divided by two and square root two divided by two. So the Y value there is square root two divided by two, that's not square root three divided by two. And then finally, for two pi divided by three, the XY values are negative one half and square root three divided by two. And there it is, there is our square root three divided by two, that's our Y value. And so the angle that we're talking about here is two pi divided by three. We look at our answer choices and that matches with answer choice. B 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/2, π] ; sin s = 1/2

Key Concepts

Unit Circle and Reference Angles

Sine Function Values and Their Symmetry

Interval Restriction and Solution Selection

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