8. what is the exact value of tan(θ) if θ = 5π/3 radians? a. -√3 b. -1 c. -√3/3 d. √3

Explanation:

Step1: Recall the tangent - angle formula

$\tan\theta=\frac{\sin\theta}{\cos\theta}$

Step2: Find the values of $\sin(\frac{5\pi}{3})$ and $\cos(\frac{5\pi}{3})$

We know that $\frac{5\pi}{3}=2\pi-\frac{\pi}{3}$. Using the unit - circle properties, $\sin(2\pi - \alpha)=-\sin\alpha$ and $\cos(2\pi-\alpha)=\cos\alpha$. So, $\sin(\frac{5\pi}{3})=-\sin(\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$ and $\cos(\frac{5\pi}{3})=\cos(\frac{\pi}{3})=\frac{1}{2}$.

Step3: Calculate $\tan(\frac{5\pi}{3})$

$\tan(\frac{5\pi}{3})=\frac{\sin(\frac{5\pi}{3})}{\cos(\frac{5\pi}{3})}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\sqrt{3}$

Answer:

a. $-\sqrt{3}$