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Find the exact value of each expression.

(a) $ \arcsin (\sin (\frac{5\pi}{4})) $

(b) $ \cos (2 \sin^{-1} (\frac{5}{13})) $

a) $-\frac{\pi}{4}$

b) $\frac{119}{169}$

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Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Okay, so we have multiple layers to take care of here. We're going to start on the inside and find the sign of five pile. Report five pi over four. Radiance is 225 degrees, so let's find the sign of 225 degrees. I'm going to draw a reference triangle in Quadrant three. Here's my 225 degree angle in standard position, and my reference triangle has a 45 degree angle in it. So this triangle is a 45 45 90 with sides of length 11 and square root toot, and I'm going to put negatives on my ones to show that I went down and I went left Now for the sign of that angle sign is always opposite over high pot news. So the inside part of what we're figuring out is negative one over square root, too. So now at this point, we have arc sine of negative one over square root, too. So let's figure that out Well, remember when we're doing an arc sine problem, were limited to arrange between negative pi over two and pi over two, so we're limited to quadrants. Four and one and where would sign be negative in quadrant four. So we're going to draw a reference triangle in quadrant four. It's sign is negative one over square root to so put negative one on the opposite and square root to on the high pot news. And there we have another 45 45 90 triangle And if you go clockwise 45 degrees, that's what we have. So we call that negative 45 degrees. Now we convert that to radiance, and our answer is negative. Pi over four radiance. So notice it did not just work to think of cancelling sign and arc sine and leaving an answer of five pi over four because five pi over four does not fall into the appropriate range. Okay, let's look at part two. So here we have the coastline of two times the Enver sign of 5/13. Remember when you look at Amber Sign of 5 13 So you want to think that that is an angle and I'm gonna name it angle Fada. So we're finding the co sign of two times data. Now that should bring to mind some trig identities. Hopefully, you learned some co sign double angle identities. There are multiple versions, and this is the one I'm choosing to use this time. So what we want to do first is work on the inside the angle Whose Sinus? 5/13. So let's make a reference triangle in Quadrant one, since it's a positive 5 13th sign is opposite overhead pot news. So we put the five on the opposite and the 13 on the high pot news. Here's our angle, Fada, and we use the Pythagorean theorem to find the length of the missing side, just in case we need it. And it's 12. Okay, so now what we're going to do is find the co sign of two times that angle. Using our identity that's going to be one minus two times a sine of the angle squared. So one minus two times well, what's the sine of the angle? Shouldn't be a surprise. Is 5/13. That's the same number we had to start with. So one minus two times 5/13 squared. So let's go ahead and square the 5/13 and we get 25 1 69th We multiply that by two and we have one minus 51 69th and then we subtract and we get 1 19 1 60 nights