The Goal

Find

The Mistake

Find the mistake:

(Roll the mouse over the math to see a hint in red)

The Correction

(Roll the mouse over the area above to see the correction in blue)

An Explanation

The notation sin^-1(x) has been misunderstood to mean 1/sin(x). That interpretation might seem to be a natural extension of notation such as sin²(x), which does mean (sin(x))², and sin^-2(x), which does mean 1/(sin(x))². However, for any function f(x), by convention when we write f^-1(x), we mean instead the inverse function for f(x). So sin^-1(x) means the inverse sine of x, that is, the function that undoes the sine function. It is not equal to 1/sin(x). The inverse sine function has the derivative shown in "The Correction".