Simplify the expression algebraically. $$5 \sin \left( \frac { \pi } { 6 } + x \right)$$
A) $\frac { 5 } { 2 }$ $( \cos x - \sqrt { 3 } \sin x )$
B) $\frac { 5 } { 2 }$ $( \cos x + \sqrt { 3 } \sin x )$
C) $\frac { 5 } { 2 }$ $( \sin x - \sqrt { 3 } \cos x )$
D) $\frac { 5 } { 2 }$ $( \cos x + \sin x )$
E) $\frac { 5 } { 2 }$ $( \sin x + \sqrt { 3 } \cos x )$

Question

Simplify the expression algebraically. $$5 \sin \left( \frac { \pi } { 6 } + x \right)$$ ​
A) $\frac { 5 } { 2 }$ $( \cos x - \sqrt { 3 } \sin x )$
B) $\frac { 5 } { 2 }$ $( \cos x + \sqrt { 3 } \sin x )$
C) $\frac { 5 } { 2 }$ $( \sin x - \sqrt { 3 } \cos x )$
D) $\frac { 5 } { 2 }$ $( \cos x + \sin x )$
E) $\frac { 5 } { 2 }$ $( \sin x + \sqrt { 3 } \cos x )$

Quizplus · Accepted Answer

Using the angle addition formula for sine, $\sin(a + b) = \sin a \cos b + \cos a \sin b$, and substituting $a = \frac{\pi}{6}$ and $b = x$, we get $\sin\left(\frac{\pi}{6} + x\right) = \sin\frac{\pi}{6}\cos x + \cos\frac{\pi}{6}\sin x$. Since $\sin\frac{\pi}{6} = \frac{1}{2}$ and $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, the expression simplifies to $\frac{1}{2}\cos x + \frac{\sqrt{3}}{2}\sin x$. Multiplying by 5 gives $\frac{5}{2}(\cos x + \sqrt{3}\sin x)$.

Simplify the Expression Algebraically 5sin⁡(π6+x)5 \sin \left( \frac { \pi } { 6 } + x \right)5sin(6π​+x)

Simplify the Expression Algebraically $5 \sin \left( \frac { \pi } { 6 } + x \right)$