Question

Multiples of \(P i\) Store any number as \(X\) in your calculator. Then enter the command \(X-\tan (X) \rightarrow X\) and press the ENTER key repeatedly until the displayed value stops changing. The result is always an integral multiple of \(\pi .\) Why is this so? [Hint: These are zeros of the sine function.]

Short answer

The operation \(X-\tan (X) \rightarrow X\) forces \(X \) to approach a value where the tangent of \(X \) is zero because of the periodicity of the tangent function. In other words, it forces \(X \) to become a multiple of \(\pi \) because tangent is periodic with period \( \pi , \) and its zeros are at integral multiples of \( \pi. \)

Step-by-step solution

Understand Tangent and Sine Functions

Tangent is a trigonometric function, defined as the ratio of the sine to the cosine of an angle (tan(x) = sin(x) / cos(x)). The function is periodic, repeating every \(\pi\) radians. The sine function is also periodic, only its period is \(2\pi.\) It's important to remember that both functions have zeros at integral multiples of \( \pi. \)

Analyze the operation

Subtracting the tangent of a number from the number itself is essentially calculating the difference between the number and its ratio of sine to cosine. As the number approaches an integral multiple of \(\pi, \) the tangent of the number approaches zero, since both sine and cosine are zero at these points.

Understand why the result is a multiple of \(\pi\)

Repeatedly performing the operation essentially forces the number to approach a value where the tangent is zero, which are the points at integral multiples of \(\pi . \) This is due to the periodicity of the tangent function. Hence, the values that cause the tangent function to go to zero, and thus stop changing the value of \(X, \) are integral multiples of \( \pi. \)