Step 1: Find $\sin\left(\frac{3\pi}{4}\right)$

The angle $\frac{3\pi}{4}$ is in the second quadrant of the unit circle. In the second quadrant, the sine function is positive.

$\sin\left(\frac{3\pi}{4}\right) = \sin\left(\pi - \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

So, the exact value of $\sin\left(\frac{3\pi}{4}\right)$ is $\frac{\sqrt{2}}{2}$.

Step 2: Find $\sin\left(\frac{19\pi}{4}\right)$ using the periodic properties of sine

The sine function has a period of $2\pi$, which means:

$\sin(\theta) = \sin\left(\theta + 2\pi n\right)$

for any integer $n$.

To simplify $\sin\left(\frac{19\pi}{4}\right)$, first, find a coterminal angle between $0$ and $2\pi$:

$\frac{19\pi}{4} \text{ can be reduced by subtracting } 2\pi \text{ until it falls within the range } [0, 2\pi).$

$\frac{19\pi}{4} - 4\pi = \frac{19\pi}{4} - \frac{16\pi}{4} = \frac{3\pi}{4}$

So, $\sin\left(\frac{19\pi}{4}\right) = \sin\left(\frac{3\pi}{4}\right)$.

Final Answer:

• $\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$
• $\sin\left(\frac{19\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Would you like more details on this explanation, or do you have any additional questions?

Here are some related questions:

1. How do you find the value of cosine at $\frac{3\pi}{4}$?
2. What is the period of the sine function?
3. What is the value of $\sin\left(\frac{7\pi}{4}\right)$?
4. How do you find the sine of negative angles?
5. What is the general formula for finding sine at any angle?

Tip: Always use the unit circle to determine the exact values of trigonometric functions at standard angles like $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{6}$.