Questions: Determine whether the statement is true or false. tan^(-1)(-1) = 3π / 4

Solution Steps

To determine whether the statement is true or false, we need to evaluate the inverse tangent (arctan) of \(-1\) and compare it to \(3\pi/4\). The arctan function returns an angle whose tangent is the given number, and it typically returns values in the range \(-\pi/2\) to \(\pi/2\). We will calculate \(\tan^{-1}(-1)\) and check if it equals \(3\pi/4\).

The inverse tangent of \(-1\), denoted as \(\tan^{-1}(-1)\), is the angle \(\theta\) such that \(\tan(\theta) = -1\). The range of the inverse tangent function is \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Therefore, \(\tan^{-1}(-1) = -\frac{\pi}{4}\).

We need to compare the value of \(\tan^{-1}(-1)\) with \(\frac{3\pi}{4}\). We have: \[ \tan^{-1}(-1) = -\frac{\pi}{4} \approx -0.7854 \] \[ \frac{3\pi}{4} \approx 2.3562 \]

Since \(-\frac{\pi}{4}\) is not equal to \(\frac{3\pi}{4}\), the statement \(\tan^{-1}(-1) = \frac{3\pi}{4}\) is false.

Final Answer