To find the terminal point determined by the angle \( t = \frac{21\pi}{4} \), we need to reduce this angle to a corresponding angle within the standard range for angles in trigonometry, typically \( [0, 2\pi) \).
- First, we can find how many complete rotations (or \( 2\pi \) cycles) are in \( \frac{21\pi}{4} \). We do this by subtracting \( 2\pi \) (which is equivalent to \( \frac{8\pi}{4} \)) until we have an angle within the range \( [0, 2\pi) \).
Starting with: \[ t = \frac{21\pi}{4} - 2\pi = \frac{21\pi}{4} - \frac{8\pi}{4} = \frac{13\pi}{4} \]
We still have a value greater than \( 2\pi \), so we subtract \( 2\pi \) again: \[ t = \frac{13\pi}{4} - 2\pi = \frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4} \]
Now \( \frac{5\pi}{4} \) is within the range \( [0, 2\pi) \).
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Now, we can find the terminal point corresponding to \( \frac{5\pi}{4} \). The angle \( \frac{5\pi}{4} \) is located in the third quadrant of the unit circle.
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The reference angle can be found by subtracting \( \pi \): \[ \text{Reference angle} = \frac{5\pi}{4} - \pi = \frac{5\pi}{4} - \frac{4\pi}{4} = \frac{\pi}{4} \]
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The coordinates (terminal point) corresponding to angles in the third quadrant can be determined using the reference angle:
- Sine is negative.
- Cosine is negative.
For the reference angle \( \frac{\pi}{4} \), the sine and cosine values are: \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
Thus, the coordinates for \( \frac{5\pi}{4} \) are: \[ \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \]
Therefore, the terminal point determined by \( t = \frac{21\pi}{4} \) is
\[ \boxed{\left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right)} \]
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