use the unit circle to find the value of sin 3pi over 4 and periodic properties of trig functions to find the value of sin 19 pi over 4. type exact answers 

### 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 \(\frac{\pi}{6}\).

Learn how to use the unit circle to find the exact values of trigonometric functions. Solve specific examples including sin(3π/4) and sin(19π/4) using the periodic properties of trigonometric functions. Suitable for high school students studying trigonometry.

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