 Remember the basic concept of an inverse function. If f of a equals b, then f inverse of b is equal to a. And very important to remember, the minus 1 in f minus 1 is not an exponent and does not mean 1 divided by f. We can apply this concept of an inverse to the trigonometric functions. So if sine of x equals y, then sine inverse of y is equal to x. And likewise, cosine x equals y gives us cosine inverse y equals x, and tangent x equals y gives us tangent inverse y equal to x. Because the trigonometric functions are periodic, we have a problem. For example, sine of pi 6 is equal to 1 half, so inverse sine of 1 half should be pi 6. But sine of 5 pi 6 is also 1 half. So could we have inverse sine of 1 half equals 5 pi 6? I don't know. Could we? Well, no, we really don't want that to happen. Since we don't want the inverse sine of a to have different values, because then it wouldn't be a function, we have to restrict the range. And so this leads us to the following definition. We define the inverse sine function, written sine inverse or arc sine, satisfies inverse sine of x equals y, where sine of y equals x, and y is between minus pi over 2 and pi over 2. So for example, let's find the inverse sine of negative square root 3 over 2. So definitions are the whole of mathematics. All else is commentary. If y is the inverse sine of negative square root 3 over 2, then the sine of y is negative square root 3 over 2. Since y has to be an angle between minus pi over 2 and pi over 2, it helps if we know the sine of the angles between minus pi over 2 and pi over 2. We see that the angle whose sine is negative square root of 3 over 2 is minus pi thirds. And so the inverse sine of negative square root 3 divided by 2 is negative pi thirds. We can define the other inverse functions in a similar fashion. So the inverse cosine function, written cosine inverse or arc cos, satisfies the relationship inverse cosine of x equals y, where cosine y equals x, and this time we restrict the output to between 0 and pi. And the inverse tangent function, inverse tangent x equals y, where tangent y equals x and y is between negative pi over 2 and pi over 2. The inverse secant function will have a range restriction between 0 and pi. And there's also an inverse cosecant and inverse cotangent function that nobody uses. So let's find arc tangent of 1, arc cosine of negative 1 half, and inverse secant of 2. So definitions are the whole of mathematics. All else is commentary. Arc tan is another way of writing the inverse tangent function. So to find arc tan of 1, well, if y is arc tan 1, then tangent of y is equal to 1. So since y has to be between minus pi over 2 and pi over 2, we need to know the values of tangent between minus pi over 2 and pi over 2. And so we see tangent is 1 at pi over 4. Arc cosine definitions are the whole of mathematics. All else is commentary. Arc cosine is another way of writing inverse cosine. And so to find arc cosine of negative 1 half, y equals arc cosine negative 1 half means the cosine of y is negative 1 half. Since y is between 0 and pi, we want the cosine values between 0 and pi. We find the angle whose cosine is negative 1 half is 2 pi thirds. Inverse secant. So again, definitions are the whole of mathematics. All else is commentary. To find the inverse secant of 2, we know that y is the inverse secant of 2 if secant of y is equal to 2. And that means we need to recall our secant values. Well, that's OK. We've memorized the values of secant. Well, nobody does that. But definitions are the whole of mathematics. All else is commentary. Secant is 1 over cosine. And we're arranging that tells us the cosine of y is equal to 1 half. And we pull in our values for cosine, and we see that cosine of y is 1 half when y is equal to pi thirds. And so we know the inverse secant of 2 is pi thirds.