find the exact value of \\( \\sin 60 ^ { \\circ } + \\sin 120 ^ { \\circ } - \\sin 240 ^ { \\circ } + \\sin…

find the exact value of \\( \\sin 60 ^ { \\circ } + \\sin 120 ^ { \\circ } - \\sin 240 ^ { \\circ } + \\sin 300 ^ { \\circ } \\).\nselect the correct choice below and fill in any answer boxes in your choice.\n\\( \\bigcirc \\) a. \\( \\sin 60 ^ { \\circ } + \\sin 120 ^ { \\circ } - \\sin 240 ^ { \\circ } + \\sin 300 ^ { \\circ } = \\)\n(type an exact answer, using radicals as needed.)\n\\( \\bigcirc \\) b. the answer is undefined.

find the exact value of \\( \\sin 60 ^ { \\circ } + \\sin 120 ^ { \\circ } - \\sin 240 ^ { \\circ } + \\sin 300 ^ { \\circ } \\).\nselect the correct choice below and fill in any answer boxes in your choice.\n\\( \\bigcirc \\) a. \\( \\sin 60 ^ { \\circ } + \\sin 120 ^ { \\circ } - \\sin 240 ^ { \\circ } + \\sin 300 ^ { \\circ } = \\)\n(type an exact answer, using radicals as needed.)\n\\( \\bigcirc \\) b. the answer is undefined.

Answer

Explanation:

Step1: Find the value of each trigonometric function

  • $\sin60^{\circ}=\frac{\sqrt{3}}{2}$
  • $\sin120^{\circ}=\sin(180^{\circ} - 60^{\circ})=\sin60^{\circ}=\frac{\sqrt{3}}{2}$
  • $\sin240^{\circ}=\sin(180^{\circ}+60^{\circ})=-\sin60^{\circ}=-\frac{\sqrt{3}}{2}$
  • $\sin300^{\circ}=\sin(360^{\circ}-60^{\circ})=-\sin60^{\circ}=-\frac{\sqrt{3}}{2}$

Step2: Substitute the values into the expression

[ \begin{align*} &\sin60^{\circ}+\sin120^{\circ}-\sin240^{\circ}+\sin300^{\circ}\ =&\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}-(-\frac{\sqrt{3}}{2})+(-\frac{\sqrt{3}}{2})\ =&\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}\ =&\sqrt{3} \end{align*} ]

Answer:

A. $\sin60^{\circ}+\sin120^{\circ}-\sin240^{\circ}+\sin300^{\circ}=\sqrt{3}$