sin(\\frac{5\\pi}{4})

Explanation:

Step1: Use the angle addition formula

We know that (\frac{5\pi}{4}=\pi+\frac{\pi}{4}). According to the formula (\sin(A + B)=\sin A\cos B+\cos A\sin B), here (A=\pi), (B = \frac{\pi}{4}). So (\sin(\frac{5\pi}{4})=\sin(\pi+\frac{\pi}{4})=\sin\pi\cos\frac{\pi}{4}+\cos\pi\sin\frac{\pi}{4})

Step2: Substitute the values of trigonometric functions

We know that (\sin\pi = 0), (\cos\pi=- 1), (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}). Substituting these values into the above formula: [ \begin{align*} \sin(\frac{5\pi}{4})&=0\times\frac{\sqrt{2}}{2}+(-1)\times\frac{\sqrt{2}}{2}\ &=-\frac{\sqrt{2}}{2} \end{align*} ]

Answer:

(-\frac{\sqrt{2}}{2})