>> This problem says the cosine of arc tangent two. Now arc tangent two is a request for the angle whose tangent is two. And this is the cosine of that angle whose tangent is two. Now it turns out that we can make this calculation even without determining the size of the angle. You see, we can do it like this, we can say okay I know that the angle whose tangent is two is an angle in a right triangle and I can actually compose that right triangle like this, and if I'm talking about the perspective of this angle, the angle whose tangent is two. Now a tangent is opposite over adjacent. So all I need to do is make sure that this fraction is equal to two. So I can make this four over two or ten over five or twelve over six, but the easy thing to do is just to make it two over one. And now I can find the length of this side and once I have the length of this side I can find the cosine of that angle, you see? Without even finding the size of the angle itself. All right, I can find the hypotenuse like this, C squared is A squared plus B squared. So C squared is two squared plus one squared. C squared then is four plus one five, C then is plus or minus the square root of five and we want the positive version, so it's the square root of five that we're after. So the square root of five is the length of the hypotenuse and now the cosine of this angle, is adjacent over hypotenuse. So it's one over the square root of five. So this becomes one over root five. And we would probably go to the trouble of rationalizing here. But the important thing is the calculation according to this request. The request here is for the angle whose tangent is two. And once we have established the situation on a right triangle, we can find the cosine of that angle. And we do it by finding the lengths of the sides within the right triangle rather than finding the sides of the angle. Here's another one, we're going to work it the same way. We're looking for the tangent of the inverse sine of negative three fourths. Now understand what this means. This is a request for an angle whose sine is negative three fourths. An angle whose sine is negative? Where does that fall? Well we know with inverse functions that for the sine the angle is either in the fourth quadrant or the first quadrant. And since the value is negative, we have a fourth quadrant angle. Ah a fourth quadrant angle, we have a fourth quadrant right triangle too. The situation looks like this. We're talking about an angle that falls in the fourth quadrant and a right triangle in the fourth quadrant and let's see, the sine of this angle is negative three fourths. Now negative three fourths. One of the either the numerator or denominator is negative, but we know that the hypotenuse is always positive. And so in this opposite over hypotenuse idea, the hypotenuse is positive. So the opposite side is negative three, the opposite side from the perspective of this angle. Okay. Now we have two sides, we just figure the third side and then we can calculate the tangent of the angle. Alright let's do that. We have C squared then is A squared plus B squared. The hypotenuse is four so this is four squared equals negative three squared plus B squared. This is sixteen, that's nine, so we have sixteen minus nine equals B squared. B squared then is equal to sixteen minus nine is seven, so B is equal to the square root of seven. And it's the positive square root because we want the distance along this positive X axis and so this is the square root of seven. Now the tangent of this angle is, let's see, opposite over adjacent, it is negative three over the square root of seven. So we right negative three over the square root of seven. And we can rationalize from here if we want to.