Sin Pi 4 5148F5

1. **State the problem:** We need to find the exact value of $\sin\left(\frac{\pi}{4}\right)$. 2. **Recall the formula and important rules:** The angle $\frac{\pi}{4}$ radians is equivalent to $45^\circ$. The sine of $45^\circ$ is a well-known value from the unit circle and special right triangles. 3. **Use the special right triangle:** A $45^\circ-45^\circ-90^\circ$ triangle has sides in the ratio $1:1:\sqrt{2}$. The sine of $45^\circ$ is the ratio of the opposite side to the hypotenuse. 4. **Calculate the sine:** $$\sin\left(\frac{\pi}{4}\right) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}.$$ 5. **Rationalize the denominator:** $$\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}.$$ 6. **Final answer:** $$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.$$