Problem 75 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan \left(-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

\(\tan \left(-\frac{\pi}{4}\right) = -1\).

Step 1: Convert the negative angle into positive

Let's convert the negative angle into its positive equivalent first. Remember that negative angles are clockwise rotations, while positive angles are counterclockwise. The positive equivalent of a negative angle can be obtained by adding \(2\pi\). However, because \(-\frac{\pi}{4}\) is equivalent to its positive counterpart +\(\frac{7\pi}{4}\), there is no need to add \(2\pi\). The reference angle is then \(+\frac{\pi}{4}\).

Step 2: Determine the value for the positive angle

The tangent of any angle in the unit circle is given by the ratio of the sine and cosine of that angle. For \(\frac{\pi}{4}\), both the sine and cosine values are \(+\frac{1}{\sqrt{2}}\) (or \(+\frac{\sqrt{2}}{2}\) when rationalized). Thus, \(\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = 1\).

Step 3: Consider the sign of the original angle

The original angle was negative, indicating that the value occurs in a quadrant where tangent is negative. Given that \(\tan(-\theta) = -\tan(\theta)\) for any angle \(\theta\), the value of \(\tan\left(-\frac{\pi}{4}\right) = - \tan\left(+\frac{\pi}{4}\right) = -1\). Thus, the tangent of \(-\frac{\pi}{4}\) is \(-1\).