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

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

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

Answer

Explanation:

Step1: Find the value of each trigonometric function

  • $\cos60^{\circ}=\frac{1}{2}$
  • $\cos120^{\circ}=\cos(180^{\circ} - 60^{\circ})=-\cos60^{\circ}=-\frac{1}{2}$
  • $\cos240^{\circ}=\cos(180^{\circ}+60^{\circ})=-\cos60^{\circ}=-\frac{1}{2}$
  • $\cos300^{\circ}=\cos(360^{\circ}-60^{\circ})=\cos60^{\circ}=\frac{1}{2}$

Step2: Substitute the values into the expression

[ \begin{align*} \cos60^{\circ}+\cos120^{\circ}-\cos240^{\circ}+\cos300^{\circ}&=\frac{1}{2}+(-\frac{1}{2})-(-\frac{1}{2})+\frac{1}{2}\ &=\frac{1}{2}-\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\ &=1 \end{align*} ]

Answer:

A. $\cos60^{\circ}+\cos120^{\circ}-\cos240^{\circ}+\cos300^{\circ}=1$