5

Find the exact value of the expression. 6251/1n 5...

Question

Find the exact value of the expression. 6251/1n 5

Find the exact value of the expression. 6251/1n 5



Answers

Find the exact value of the expression, if it is defined.
$\sin \left(\sin ^{-1} 5\right)$

Okay, here. We have an interesting question. We have the sign of the inverse co sign of 3/5. Now, the interesting thing is that we know that signing coastline on Lee worked with not only but in our case, we're looking at right triangles. So we have a right triangle like this. We know that because this is co sign this is 3/5. So 3/5 overlook. We would be looking for this angle and we find the sign of that angle. Now note. This is three. This is five. This is an interesting triangle called a three 45 triangle. If you were to use the Pythagorean there, nine plus 16 equals 25. So this has to be four squared or just for because of that. If we do the if inverse co sign of negative one for 3/5, we get data, so we get signed of fada. But because just really matter what they does in this case, because we know that data sign is opposite overhype Otterness. We get 4/5. So because of that, our answer is 4/5 and that's it

So here we're looking for the tangents of the inverse sign of 4/5. Now 4/5 is an interesting number set, because if we're looking at it before looking at a right triangle and we have fate over here, that means this would before and this would be five. So by finding the inverse sign we established with David's now after that, we would find the tangent of this a fatal whatever they don't might be. So it's actually kind of easier to figure out what this third site it's. So if we use the pipe that you're in there and we can do 25 minus 16 because five square minus four square is equal to whatever this site is, and that's equal to nine, you take the square tonight and we get three. This is a right triangle, right? Consistent right triangle called the 345 triangle. And it's always gonna be looked like this because of that. What we can do. Once we found theta fighting the inverse I in of for over five, we can use the tensions, which is opposite over Jason. So we get 4/3, and if we get for over three, that is our answer. Based on the trip

So co sign of sine inverse of 4/5 that will be equal to 0.6.

In this problem. We're looking to simplify the expression inverse sign of sign of five pi over six. And for some problems we've done in this section, we've used this property here. If f in gear in versus then f of g of X equals X. So it would be nice if we could apply that to this problem and just cancel our inverse functions and give. Our answer is five pi over six. However, there is a problem with that because if you recall the function, why equals inverse Sign of X has a restricted range and the range is from negative pi over to two pi over two, which means that our answer has to fall in that range and five pi over six is not in that range. So here's what we'll do. Instead, we'll work on this from the inside out, starting with finding the value of the sign of five pi over six. So let me go ahead and draw an angle. That's five pi over six radiance that would be 150 degrees. So this is 1 50 So here's my reference triangle with a 30 degree angle so we can label the sides of the 30 60 90 triangle. And then from here we find the angle Whose sign From here we find the sign of five pi over six. So the sign of five pi over six would be opposite over high pot news, and that would be one half. So we're going to replace this expression with one half. So now the problem is find the inverse sign of one half. Okay, we are going to use quadrant one when Enver Sign has a positive argument again, Sign is opposite over high pot news. So here we have another 30 60 90 triangle. But it's in quadrant one with a 30 degree reference angle. So if we're answering in degrees are answer would be 30 degrees. If we're answering and radiance, our answer would be pi over six.


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