Question

Explain why \(\tan \pi=0\) does not imply that \(\arctan 0=\pi\).

Short answer

The reason why \(\tan \pi=0\) doesn't imply that \(\arctan 0=\pi\) is because the \(\tan\) function has a period of \(\pi\) which is not inherited by its inverse, the \(\arctan\) function. Furthermore, the \(\arctan\) function's range is \(-\pi/2\) to \(+\pi/2\), which further justifies why \(\arctan 0\) does not equal to \(\pi\).

Step-by-step solution

Reasoning About the Periodic Behavior of the Tangent Function

In mathematics, the tangent function is an example of a periodic function, specifically with a period of \(\pi\) - it repeats its values at every interval of \(\pi\). This is the reason why \(\tan(\pi) = \tan(0) = 0\). However, this does not confer the same property to its inverse function, the arctangent.

Understanding the Inverse Trigonometric Function

The inverse trigonometric function 'arctan' or \(\tan^{-1}\) is used to obtain an angle from the tangent of that angle. For \(\arctan(0)\), the angle that gives a tangent of 0 is 0, not \(\pi\). Inverse trigonometric functions are not periodic and hence don't inherit the periodic behaviors of their respective trigonometric functions.

The Inequality that Bounds the Inverse Tangent Function

Moreover, the range of the arctan function is between \(-\pi/2\) and \(+\pi/2\). This means the output of the function will always lie within this range. Therefore, the function \(\arctan(0)\) does not equal \(\pi\).