 When working with the inverse trigonometric functions, it's important to keep the range in mind. So remember arc sine and arc tangent x always gives an angle in quadrant 1 or 4, while arc cosine x always gives a quadrant 1 or 2 angle. For example, let's try to find arc sine of sine of pi thirds. Now, since arc sine and sine are inverse functions, they might cancel. If f inverse is the inverse of f, then f inverse of f of x is x, provided that x is in the range of f inverse. And so it's important to remember that arc sine of x returns an angle in the interval between negative pi halves and positive pi halves. So that means we need to verify that our output is actually in the correct interval. And since pi thirds is between negative pi halves and pi halves, then the arc sine of the sine of pi thirds is in fact pi thirds. However, rather than trying to determine afterwards if the output is in the correct interval, we can take the following approach, draw a picture, and then find the angle in the appropriate quadrant that gives the same trigonometric value. So again, if we want to find the arc sine of the sine of pi thirds, we draw a picture representing the sine of pi thirds. So we'll draw a unit circle, we'll rotate through the angle of pi thirds, and then since we're looking at the sine value, we'll drop a line down to the x-axis. Now, since arc sine returns an angle in quadrant one or four, we look for a quadrant one angle with the same sine value. But since we're already in quadrant one, this will be the same angle we started with, and so arc sine of sine of pi thirds is pi thirds. So for example, suppose you want to find the arc sine of the sine of five pi fourths. And here's the quick wrong answer. Since sine x and arc sine x are inverse functions, they cancel each other, and arc sine of sine of five pi fourths is five pi fourths. And this illustrates the idea that how you speak influences how you think. Try to avoid the word cancel because it leads to thinking like this, which in this case is incorrect. Instead, let's draw a picture. So we'll begin with our unit circle and rotate through an angle of five pi fourths. And since sine is the vertical distance of the point, let's drop the line to the x-axis. Now, since arc sine of x must be an angle in quadrant one or four, we need the angle in this quadrant that gives us the same sine value. And this is going to be a reflection across the y-axis. So we need to find the reference angle, which will be pi fourths. So the angle in the fourth quadrant will also be pi fourths, but since we will have rotated clockwise, the arc sine will be negative pi fourths. Or we could have something like the arc cosine of the cosine of negative two. Now remember, if no units are specified, assume that our angle is measured in radians. So we'll draw a picture. Since the angle measure is negative, we turn clockwise. And since two is less than pi, we're going to turn slightly less than halfway around. Now, since our cosine x must be in quadrant one or two, we need to find the angle with the same cosine value. And that's our x-coordinate. So this is going to be a reflection across the x-axis. So the angle measure will be the same, but since we're now measuring it counterclockwise, it's going to be positive. And so the arc cosine of the cosine of negative two will be two.