evaluate. write your answer in simplified, rationalized form. do not round. \n\n$\\cos\\left(\\frac{\\pi}{6}\…

Explanation:

Step1: Recall the cosine value of special angles

We know that for the unit - circle, the cosine of an angle $\theta$ in standard position ($\theta$ measured from the positive $x$ - axis) is given by the $x$ - coordinate of the point on the unit - circle corresponding to the angle $\theta$. The angle $\theta=\frac{\pi}{6}$ (or $30^{\circ}$) is a special angle. On the unit - circle, for the angle $\theta = \frac{\pi}{6}$, the coordinates of the corresponding point are $(\cos\frac{\pi}{6},\sin\frac{\pi}{6})$. We know that $\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$ and $\sin\frac{\pi}{6}=\frac{1}{2}$ from the properties of a $30 - 60-90$ triangle (where in a right - triangle with angles $30^{\circ}-60^{\circ}-90^{\circ}$, if the side opposite the $30^{\circ}$ angle is $a$, the side opposite the $60^{\circ}$ angle is $\sqrt{3}a$ and the hypotenuse is $2a$, and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$ for an acute angle $\theta$ in a right - triangle).

Answer:

$\frac{\sqrt{3}}{2}$