find the terminal point on the unit circle determined by $\\frac{5\\pi}{4}$ radians. use exact values, not…

Explanation:

Step1: Determine the reference angle

The reference angle $\bar{\theta}$ for $\theta=\frac{5\pi}{4}$ is $\bar{\theta}=\frac{5\pi}{4}-\pi=\frac{\pi}{4}$.

Step2: Find the coordinates using trigonometric functions

For a unit - circle, the terminal point $(x,y)$ is given by $x = \cos\theta$ and $y=\sin\theta$. Since $\theta=\frac{5\pi}{4}$ is in the third quadrant, where $\cos\theta<0$ and $\sin\theta<0$. We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. So, $x=\cos\frac{5\pi}{4}=-\cos\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$ and $y = \sin\frac{5\pi}{4}=-\sin\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$.

Answer:

$(x,y)=\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$