Question

question given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\( \cos\left( -30^{\circ} \
ight) \\)

Explanation:

Step1: Recall the cosine even - function property

The cosine function is an even function, which means that $\cos(-\theta)=\cos(\theta)$ for any angle $\theta$. So, for $\cos(- 30^{\circ})$, we can rewrite it as $\cos(30^{\circ})$.

Step2: Recall the exact value of $\cos(30^{\circ})$

From the special right - triangle (a 30 - 60 - 90 triangle) or the unit - circle definition, we know that in a 30 - 60 - 90 triangle, if the side opposite the $30^{\circ}$ angle has length $x$, the hypotenuse has length $2x$, and the side opposite the $60^{\circ}$ angle (adjacent to the $30^{\circ}$ angle) has length $x\sqrt{3}$. Using the definition of cosine as $\cos(\alpha)=\frac{\text{adjacent}}{\text{hypotenuse}}$, for $\alpha = 30^{\circ}$, $\cos(30^{\circ})=\frac{\sqrt{3}}{2}$. Since $\cos(-30^{\circ})=\cos(30^{\circ})$, we have $\cos(-30^{\circ})=\frac{\sqrt{3}}{2}$.

Answer:

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