Question

standard position. sketch the angle -\frac{7\pi}{4}-4\pi radians.

Explanation:

Step1: Recall angle - reduction property

Since adding or subtracting multiples of \(2\pi\) to an angle does not change its position. We know that \(- 4\pi=-2\times2\pi\) is a full - rotation multiple. So we can focus on the angle \(-\frac{7\pi}{4}\).

Step2: Find the equivalent positive angle

We know that \(-\frac{7\pi}{4}+2\pi=\frac{-7\pi + 8\pi}{4}=\frac{\pi}{4}\). A negative angle is measured clock - wise. The angle \(-\frac{7\pi}{4}\) radians is equivalent to a clock - wise rotation of \(\frac{7\pi}{4}\) radians. Since \(\frac{7\pi}{4}\) radians is \(\frac{8\pi-\pi}{4}\), it is the same as a clock - wise rotation of \(2\pi-\frac{\pi}{4}\) radians. Starting from the positive \(x\) - axis (\((1,0)\)) and rotating clock - wise by \(\frac{7\pi}{4}\) radians, we end up in the first quadrant, \(\frac{\pi}{4}\) radians (or \(45^{\circ}\)) away from the positive \(x\) - axis.

Answer:

To sketch the angle \(-\frac{7\pi}{4}-4\pi\) radians, start at the positive \(x\) - axis \((1,0)\) and make a clock - wise rotation of \(\frac{7\pi}{4}\) radians. The terminal side of the angle will be in the first quadrant, making an angle of \(\frac{\pi}{4}\) radians (or \(45^{\circ}\)) with the positive \(x\) - axis. Since \(-4\pi\) is just two full clock - wise rotations, it does not change the position of the terminal side of \(-\frac{7\pi}{4}\) radians.