Given:
There are given that the cos function:
[tex]cos210^{\circ}=-\frac{\sqrt{3}}{2}[/tex]Explanation:
To find the value, first, we need to use the half-angle formula:
So,
From the half-angle formula:
[tex]cos(\frac{\theta}{2})=\pm\sqrt{\frac{1+cos\theta}{2}}[/tex]Then,
Since 105 degrees is the 2nd quadrant so cosine is negative
Then,
By the formula:
[tex]\begin{gathered} cos(105^{\circ})=cos(\frac{210^{\circ}}{2}) \\ =-\sqrt{\frac{1+cos(210)}{2}} \end{gathered}[/tex]Then,
Put the value of cos210 degrees into the above function:
So,
[tex]\begin{gathered} cos(105^{\circ})=-\sqrt{\frac{1+cos(210)}{2}} \\ cos(105^{\operatorname{\circ}})=-\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}} \\ cos(105^{\circ})=-\sqrt{\frac{2-\sqrt{3}}{4}} \\ cos(105^{\circ})=-\frac{\sqrt{2-\sqrt{3}}}{2} \end{gathered}[/tex]Final answer:
Hence, the value of the cos(105) is shown below:
[tex]cos(105^{\operatorname{\circ}})=-\frac{\sqrt{2-\sqrt{3}}}{2}[/tex]