Question

what is \\( \sin \left( - \frac { 3 \pi } { 4 } \
ight) ? \\)

Explanation:

Step1: Use the property of sine function

The sine function is an odd function, so \(\sin(-x)=-\sin(x)\). Then \(\sin(-\frac{3\pi}{4})=-\sin(\frac{3\pi}{4})\).

Step2: Find the value of \(\sin(\frac{3\pi}{4})\)

We know that \(\frac{3\pi}{4}=\pi - \frac{\pi}{4}\), and \(\sin(\pi - \alpha)=\sin\alpha\). So \(\sin(\frac{3\pi}{4})=\sin(\pi - \frac{\pi}{4})=\sin(\frac{\pi}{4})\). And \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\).

Step3: Calculate \(\sin(-\frac{3\pi}{4})\)

Since \(\sin(-\frac{3\pi}{4})=-\sin(\frac{3\pi}{4})\) and \(\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}\), then \(\sin(-\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}\).

Answer:

The number in the numerator of the green - boxed part is \(2\), and the number in the denominator of the grey - boxed part is \(2\).