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Numerade Educator

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Problem 72 Hard Difficulty

Simplify the expression.

$ \sin (2 \arccos x) $

Answer

$2 x \sqrt{1-x^{2}}, x \in[0, \pi]$

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Video Transcript

all right, we're going to find the sign of two times the art co sign of X. First of all, remember your double angle identities. The sign of two times and angle equals two times the sine of the angle times the coastline of the angle. Okay, our coastline X means the angle. Whose son who's co sign is X, So I'm going to call it data. So we're finding the sign of two data. Now the angle. Whose coastline is X? What would that look like? There are two possibilities. One possibility is that the angle is in quadrant one. Another possibility is that the angle is in Quadrant two. It's terminal side in quadrant two, so let's look at both cases. If the angle is in quadrant one in its coastline is X, then we could put X on the adjacent and one on the high pot. News because X over one is X, and if the angle is in quadrant to weaken, do the same thing X on the adjacent one. On the hip hop news, we still have X over one is X. Now let's find the link of the opposite because we're going to need that in order to find the sign. So for now it's called the opposite Why? And it's going to be Why on the other side as well. So now we can use the Pythagorean theorem, and we have why squared Plus X squared equals one squared. So why squared equals one minus X squared? So why is the square root of one minus X squared? Okay, now we can work with this formula here to find a sign of two times the angle we want two times the sine of the angle times the coastline of the angle. What's the sine of the angle opposite over high pot news? So that would be square root, one minus X squared over one. And what's the coastline of the angle adjacent over high pot news. So that would be X over one. So now we can simplify this, and we have two times x times the square root of one minus X squared, and then just keep in mind that the inverse co sign or ARC co sign as we're calling it here function is only defined for X between zero and pie, so we might want to make a note of that. X is in the interval from zero to pi