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.

[0, 2π) ; sin s = -√3 / 2

Accepted Answer

Welcome back. I am so glad you're here. We're told, considering the interval from 0 to 2 pi find the exact values of P. Then we are told that the sign of P equals negative one half. Our answer choices are answer choice. A pi divided by six and five pi divided by six. Answer choice B seven pi divided by six and 11 pi divided by six answer choice C pi divided by three and two pi divided by three and answer choice D five pi divided by six and 11 pi divided by six. So what do we know here? We're told that our interval is from 0 to 2 pi that is a full rotation around the unit circle. I'm going to draw that real quick. We have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle over that we have our unit circle where every point on the circle is exactly one unit away from the origin. You can imagine that I drew that and then we know that our P could be anywhere on there except we have another clue to narrow it down a bit we recall from previous lessons that the sign of an angle on the unit circle, we'll call that s is equal to the Y value. So here our why value is negative? It's a negative one half. Where are the Y values negative? That's in the 3rd and 4th quadrants. So the third quadrant that's down into the left of the origin and the fourth quadrant is down into the right of the origin. So that narrows it down a little bit. Now we just have to find where the Y values equal negative one half. So I'm going to draw some of the details for the unit circle. Definitely check out the figure of the unit circle in your textbook. For a more detailed version, I'm going to start in the third quadrant closer to the negative part of the X axis than the negative part of the Y axis. We have the angle seven pi divided by six. It's xy coordinates are negative square root three divided by two, negative one half. Hey look there it is already we have our negative one half for our Y value. So one of the values for P is seven pie divided by six. Now let's pop over to the fourth quadrant and we'll take a look at some of the angles there going to start closer to the positive part of the X axis than the negative part of the Y axis. The first angle that we have there is 11 pi divided by six. Its xy coordinates are positive square three divided by two and negative one half. There is our other negative one half or our Y value. So P's second value where the sine of P equals negative one half is 11 pi divided by six. And that's within the interval from 0 to 2. Pi we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

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

[0, 2π) ; sin s = -√3 / 2

Key Concepts

Unit Circle

Sine Function

Reference Angles

Watch next