Use the given information to find each of the following.

Question

cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given trigonometric expression using the half angle identity. Our given trigonometric expression is the cosine of X divided by two. It has a couple of conditions we're told if the sign of X is equal to negative 5/13 and where X is greater than 180 degrees but less than 270 degrees. Our answer choices are answer choice. A five square root 26 divided by 26. Answer choice B negative square root 26 divided by 26. Answer choice C negative five square root 26 divided by 26 and answer choice D square root 26 divided by 26. All right. So using the half angle identity for cosine, we recall from previous lessons. That's the cosine of A divided by two equals plus or minus the square root of. And then inside that radical is a fraction in the numerator, we have one plus the cosine of A and that's divided by two. Comparing what we're given with the trigonometric half angle identity. We can say that our a value is going to be X for this one. So we can set our given expression cosine of X divided by two equal to what the cosine of A divided by two is equal to plus or minus the square root of in the numerator will have one plus the cosine of. And then instead of a, that's X all divided by two. And that's great except what is the cosine of X. Well, we know what the sine of X is. The sine of X is negative 5/13. Can we get from sine to cosine? Yes, we can. And there are several ways to do it. If you recall from previous lessons. If we have the sine of an angle, we'll say theta that is equal to the Y value divided by the R value. Here, we can see the Y value in the numerator is the negative five and the R value in the denominator is 13. That's always going to be positive because that is a distance. So Y equals negative five and R equals 13 for the cosine of theta that's equal to X divided by R. So we know R in the denominators 13, we just need to figure out X in the numerator. So we can take a moment and solve for X because we recall that R is equal to the square root of X squared plus Y squared. So I'm going to move this to a different part of the screen for a moment and take the time to solve for X. So R is 13 equals the square root of X. It's unknown. So X squared plus Y is negative five squared and bring this down 13 equals the square root of X squared plus negative five squared is 25. Now, we can square both sides. And on the left 13 squared is 169 that's equal to the square rut of X squared plus 25 squared is just X squared plus 25. Now, we can subtract 25 from both sides. On the left, 169 minus 25 is 144. That equals X squared. We'll take the square rut of both sides and the square root of 144 is plus or minus 12. So X equals plus or minus 12. But which one is it? So if we bring this back to where we were before, if we know that our X value is in between 180 degrees and 270 degrees. Where are we? We'll draw a quick sketch of a coordinate plane. We have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle 180 degrees is the negative part of the X axis. 270 degrees is the negative part of the Y axis. So if X is in between those X is in quadrant three, in quadrant three, the cosine function is negative. So we can disregard that positive for our positive 12 option. And it's just going to be a negative 12 for our numerator. And now we have that cosine of X is equal to negative 12/13. So we can plug that in. We've got the cosine of X divided by two equals plus or minus the square root of in the numerator. That's now one plus. And instead of cosine of X, the exact value of that is negative 12/13. And that's all divided by two. Simplifying in the numerator, one plus negative 12/13 is one minus 12/13 which is 1/13. So this is plus or minus the square root of 1/13 divided by two, 1/13 divided by two is 1 26 1 26th. So this is equal to plus or minus the square root of 1 26th, which looks lovely. But we need to rationalize that denominator. So we're going to multiply the numerator and the denominator by the square root of 26. And so this equals plus or minus in the numerator, one, multiple, the square root of one multiplied by the square root of 26 is going to be the square root of 26 divided by, in the denominator. The square root of 26 multiplied by itself is just 26. This looks so lovely, but we don't know if it's plus or minus. So let's figure that part out we are talking about the angle X divided by two, we know that X is in between 180 degrees and 270 degrees. To find out where X divided by two is we're going to divide all three of those parts by two. And so if we take 180 degrees divided by two, that's 90 degrees and X divided by two is greater than that, but less than 270 degrees divided by two, that's 135 degrees. So where are we? 90 degrees is the positive part of the X or excuse me, of the Y axis and 135 degrees is in quadrant two. So in between those two, we're talking about quadrant two and cosine is going to be still negative in quadrant two. So we can again disregard that positive answer. And now we know that the cosine of X divided by two. When the sine of X equals negative 5/13 the cosine of X divided by two is going to be equal to a pot or excuse me, a negative, we disregarded the positive. It's a negative square root 26 divided by 26. And that's it. Well done. We look at our answer choices and this matches with answer choice. B we'll catch on the next one.

Use the given information to find each of the following.
cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°

Key Concepts

Trigonometric Functions

Angle Properties in Quadrants

Half-Angle Formula

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