The problem asks for the point $(x, y)$ on the unit circle that corresponds to $t = \frac{-5\pi}{4}$.

Solution:

The coordinates of a point on the unit circle for an angle $t$ are given by: $(x, y) = (\cos t, \sin t)$

Step 1: Convert $t$ to a standard angle

Since angles on the unit circle are typically measured in radians from $0$ to $2\pi$, we find the equivalent positive angle:

$\frac{-5\pi}{4} + 2\pi = \frac{-5\pi}{4} + \frac{8\pi}{4} = \frac{3\pi}{4}$

Thus, $\frac{-5\pi}{4}$ corresponds to $\frac{3\pi}{4}$.

Step 2: Find $\cos$ and $\sin$

From unit circle values:

$\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}, \quad \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}$

Thus, the point is:

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

Final Answer:

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

Would you like a deeper explanation or more practice problems?

Related Questions:

1. How do you find coterminal angles for a given radian measure?
2. What is the general formula for finding points on the unit circle?
3. How do sine and cosine values change in different quadrants?
4. How do you convert between degrees and radians?
5. What is the significance of the unit circle in trigonometry?

Tip:

To find equivalent angles, always add or subtract multiples of $2\pi$ until you get an angle within $[0, 2\pi]$.