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Numerade Educator

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Problem 66 Hard Difficulty

Find the exact value of each expression.

(a) $ \sin^{-1} (\frac{-1}{\sqrt{2}}) $
(b) $ \cos^{-1} (\frac{\sqrt{3}}{2}) $

Answer

a) See solution
b) $\frac{\pi}{6}$

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Video Transcript

when you see an expression like this inverse sign of negative one over square, we to. Here's what you want to think it's asking you to find the angle. Always with inverse trig, you're finding the angle. Who's sign is negative one over a square root to okay, find the angle. Who sign is negative one over square were to remember that your outputs your angles have to fall between negative pi over two and positive pi over two for inverse sign. And if you have a negative value for the sign, then you must be in Quadrant four. If your choices are just quadrants one and quarter for you know, quadrant one is always positive, so you must be in quadrant for so draw reference triangle. We're looking for the angle, the reference angle, and we know the sign is negative, one over square root to so we can put negative one on the opposite and square it to on the high pot. News. Hopefully, at this point, we recognize that we have a special right triangle with sides of length one and one and high pot new square root to, and that is a 45 45 90 triangle, so we have a 45 degree angle in the triangle. Since we went clockwise, we're gonna call that negative 45 degrees. And because we need to answer in radiance, we're going to change that to negative pi over four radiance. Okay? Similarly, if you have inverse co sign of square root 3/2, here's what you want to think. It's asking you to find the co sign or skinny. Find the angle. Always finding an angle with inverse trig. Find the angle whose co sign is square and 3/2. And remember your outputs for inverse coastline have to fall between zero and pie. That would be quadrant one in quadrant two, and if you have a positive value, it's quiet one. So let's draw a reference triangle. We know co sign is adjacent over high pot news, so we're going to put square root three on the adjacent and two on the high pot news. Now we should recognize that as a 30 60 90 special right triangle with a 30 degree reference angle. So we're angle is 30 degrees. But let's convert that into radiance. It's pi over six