find the exact value of the expression. cos(arctan(√3))

Explanation:

Step1: Let $\theta=\arctan(\sqrt{3})$

This means $\tan\theta = \sqrt{3}$ and $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$.

Step2: Recall the right - triangle relationship

Since $\tan\theta=\frac{y}{x}=\sqrt{3}=\frac{\sqrt{3}}{1}$, we can consider a right - triangle where $y = \sqrt{3}$ and $x = 1$. Then, by the Pythagorean theorem $r=\sqrt{x^{2}+y^{2}}=\sqrt{1 + 3}=2$.

Step3: Find $\cos\theta$

We know that $\cos\theta=\frac{x}{r}$. Substituting $x = 1$ and $r = 2$, we get $\cos\theta=\frac{1}{2}$.

Answer:

$\frac{1}{2}$