4) \\( \\sin \\frac { 7 x } { 6 } \\) a) \\( - \\frac { 1 } { 2 } \\) b) \\( \\frac { 1 } { 2 } \\) c) \\(…
4) \\( \\sin \\frac { 7 x } { 6 } \\) a) \\( - \\frac { 1 } { 2 } \\) b) \\( \\frac { 1 } { 2 } \\) c) \\( - \\frac { \\sqrt { 3 } } { 2 } \\) d) \\( \\frac { \\sqrt { 3 } } { 2 } \\)
Answer
Explanation:
Step1: Convert angle to standard position
$\sin\frac{7\pi}{6}=\sin(\pi+\frac{\pi}{6})$
Step2: Use sine addition formula
According to $\sin(A + B)=\sin A\cos B+\cos A\sin B$, here $A = \pi$, $B=\frac{\pi}{6}$. $\sin(\pi+\frac{\pi}{6})=\sin\pi\cos\frac{\pi}{6}+\cos\pi\sin\frac{\pi}{6}$ Since $\sin\pi = 0$, $\cos\pi=- 1$, $\sin\frac{\pi}{6}=\frac{1}{2}$, $\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$ So $\sin(\pi+\frac{\pi}{6})=0\times\frac{\sqrt{3}}{2}+(-1)\times\frac{1}{2}=-\frac{1}{2}$
Answer:
A. $-\frac{1}{2}$