>> Now that we have information about inverse trigonometric functions, let's work some problems. This problem says the inverse cosine of 1 over the square root of 2, and let's understand that this is a request for the angle whose cosine is 1 over the square root of 2, is this a cosine value? Yes, it's vaguely familiar. Let's just say we're not using the calculator here otherwise, we could just punch this and make the calculation. So we want to do it by hand and if this is one of those familiar right triangle situations that we've had in the past and we can't recall it, then what we do is we slide over to the side and we sketch a couple of right triangles. Now one of the right triangles you want to sketch right quick is the one with a 60-degree angle and a 30-degree angle. And remember, in this right triangle, the short side is opposite the small angle so this is the 1, 2 square root of 3 situation. The other triangle you want to sketch is the one with 45-degree angles for the acute angles. We only need information about one of them because they're kind of the same. But at any rate, it's the 1, 1 square root of 2 situation. Now from the perspective of what angle, is it the case that the cosine of the angle is 1 over the square root of 2. That's really, what is the request here. Oh, it's this one because cosine is 1 over the square root of 2. Oh, it's 45 degrees so we come back over, 45 degrees and that's our answer. Now if we want to we can convert this into radian measure or we could have thought of these triangles as being measured in radians as well. Arc tangent, the square root of 3, means that we're looking for the angle whose tangent is the square root of 3. We look over here, oh the square root of 3, there it is and let's see, tangent is opposite over adjacent so the square root of 3 over 1, oh the perspective is from this angle so the angle is 60 degrees. Once again, we could convert to radians very easily. Arc sine, square root of 3 over 2, ah, the one that involves square root of 3 is this one. Now from the perspective of which angle, we have the sine as the square root of 3 over 2. Now that's opposite over hypotenuse, square root of 3over2, oh the perspective is again, from this 60 degree angle. Let's use the calculator to find these. Arc cosine negative .647, arc cosine. Now that means we want the angle whose cosine is this negative value. Now we know probably at the beginning that cosine, the cosine of angles is positive in the 1st and 4th quadrants but negative in 2nd and 3rd and we know that in the world of inverse functions, that we're talking about angles that appear only in the 1st and 2nd quadrant for the inverse cosine. All right? And therefore, this is a 2nd, going to be a 2nd quadrant angle, okay, because it's negative. Okay. Now to use the calculator we do this, we press 2nd and then cosine, the 2nd function on all of the trig functions are the inverses of them. So this is the inverse of the cosine. Okay, so it's like arc cosine or inverse cosine of, and then you just enter the number and press enter and, depending on the mode of your calculator, it will give you the answer in degrees or in radians. If you're in degree mode, the angle is approximately 130.3 degrees. I have it written off camera here, I'm reading it and writing it. I can't calculate these things on my own, you know. And if you're in radian mode, the calculator says that it's 2.27.