If we know tangent alpha equals 15 eighths and alpha lies in quadrant 3, we're going to put in a right triangle in quadrant 3, so opposite sides 15 adjacent 8. Using the Pythagorean identity, we can find the hypotenuse is 17. If alpha is in the third quadrant, we know that alpha has got to be in between 180 and 270. So alpha over two has got to be in between 90 and 1:35. So if we think about in between 90 and 135, now we're talking about in between 90 and 1:35. So we'd be in the second quadrant. And we know in the second quadrant Sine's positive, Cosine's negative, and tangent is negative. So now it's just a matter of actually finding what those are. Sine alpha over two is going to be the positive square root 1 minus cosine alpha. Cosine alpha in this case is going to be -8 seventeenths all over two. So when we compute this, we're going to have 17 + 8 all over. 3417 + 8 is 25 S sqrt 25 over 34. We're not going to leave a radical in the bottom, so we're going to rationalize sqrt 35 / sqrt 35 and sqrt 25 was 5 sqrt 34 / 34. When we look at cosine alpha over two, cosine alpha over two is going to be one plus cosine alpha -8 seventeenth all over 2. So now we get 17 -, 8 all over 34. The square root of that 17 -, 8's going to give us 9. So sqrt 9 / sqrt 34. Rationalize that sqrt 34 and sqrt 34. So we're going to get 3 sqrt 34 / 34. Now remember that this is actually in the 2nd quadrant in between 90 and 1:35. So these are all going to be negatives because we know that cosine in the second quadrant is a negative. To get the tangent, the tangent we can just do as sine over cosine. So tangent of alpha over 25 root 34 / 34 / -3 root 34 / 34 so -5 thirds.