In which quadrant does the terminal side of a \( -\frac{5 \pi}{4} \) radian angle in standard position lie? Quadrant I Quadrant II Quadrant III

To find where the terminal side of the angle \( -\frac{5 \pi}{4} \) radians lies, we first convert the negative angle into a positive equivalent. This is done by adding \( 2\pi \) (or \( \frac{8\pi}{4} \)) to \( -\frac{5\pi}{4} \), which gives us \( \frac{3\pi}{4} \). The angle \( \frac{3\pi}{4} \) radians is in the second quadrant where angles range from \( \frac{\pi}{2} \) to \( \pi \). In practical applications, understanding angles and their positions is essential for fields like engineering and navigation. For instance, when designing a ramp, the angle of elevation must be accurately measured to ensure that vehicles can ascend safely. Knowing which quadrant an angle lies in can help in determining the proper sine and cosine values crucial for calculations.