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Inverse Functions

Part 2: Inverting the Sine Function

In this part we consider what it means to invert the sine function. No doubt you have used the key on your calculator labeled sin-1 -- and you may know exactly what it does. But the way in which values of the inverse sine are calculated may still be something of a mystery -- because it is essentially a calculus problem. We won't get into exactly what your calculator does, but we will use calculus to find a way to evaluate the inverse sine using just arithmetic.

First we note that there is a complication with trigonometric functions that did not arise with exponential and logarithmic functions. The following figure shows two periods of the sine function (red) and the result (blue) of interchanging x and y in the equation y = sin x.

Sine

The problem is that the blue curve is obviously not the graph of a function. In order to have an invertible function that takes all the values of the sine function, we restrict the domain of the sine to the interval from - pi/2 to pi/2. This restricted sine and its inverse are shown in the next figure.

Sine and Inverse

  1. What is the domain of the inverse sine function? What is the range?

For now, we propose a function that appears to have the right sort of graph -- in the next part we take up a technique for predicting formulas of this sort. Our proposed formula is

Arcsine integral

The graph of this function is shown in the following figure -- but of course the graph is not enough evidence to claim this really is the inverse function.

Arcsine integral

  1. What is the domain of the function F?

  2. Evaluate F(x) for various values of x. In particular, evaluate F(1). [Note: The integral defining F(1) is called "improper" because the integrand is discontinuous at x = 1. Most computer algebra systems will know how to evaluate this integral -- symbolically, not numerically.]

  3. In your worksheet, calculate F(sin x) and sin(F(x)) for various values of x. Record your results.

  4. Graph the functions F(sin x) and sin(F(x)). Explain what you see. In particular, what are the domains of these two composite functions? [Warning: Your computer algebra system may not know the domains of these functions and may do something strange. You are responsible for a correct mathematical answer even if the computer result is misleading. You don't have to explain what the CAS is doing -- just explain the mathematics.]

  5. Is the function F the inverse of the sine function restricted to the interval [-pi/2,pi/2]? Why or why not?

For each value of x, the function value F(x) is a definite integral of an algebraic function, that is, one whose values may be calculated by arithmetic operations and extraction of roots. We know that definite integrals can be evaluated numerically to high accuracy (recall the Accumulation module), using only arithmetic operations and values of the integrand. Thus, in principle, your calculator could be evaluating integrals when you press the sin-1 key. However, the actual wiring uses a much faster numerical algorithm.

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