Using the unit circle calculate the sine and cosine values of the angle below. Find sin(t) and cos(t) when t=(2pi )/(3) radians.

To find the sine and cosine values of the angle \( t = \frac{2\pi}{3} \) radians using the unit circle, we can follow these steps:<br /><br />1. **Locate the angle on the unit circle**: <br /> - The angle \( t = \frac{2\pi}{3} \) radians is in the second quadrant because it is greater than \( \frac{\pi}{2} \) but less than \( \pi \).<br /><br />2. **Determine the reference angle**:<br /> - The reference angle for \( t = \frac{2\pi}{3} \) is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \).<br /><br />3. **Use the unit circle values for the reference angle**:<br /> - For \( \frac{\pi}{3} \), we know from the unit circle that:<br /> \[<br /> \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}<br /> \]<br /><br />4. **Determine the signs of sine and cosine in the second quadrant**:<br /> - In the second quadrant, sine is positive and cosine is negative.<br /><br />Therefore, we have:<br />\[<br />\sin\left(\frac{2\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}<br />\]<br />\[<br />\cos\left(\frac{2\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}<br />\]<br /><br />So, the values are:<br />\[<br />\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}<br />\]<br />\[<br />\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}<br />\]