Question

Find the exact value of csc theta if tan theta = sqrt3 and the terminal side of theta is in Quadrant III.

Answer 1

Answer:

3rd option

Step-by-step explanation:

Using the identities

cot x = $\frac{1}{tanx}$

csc² x = 1 + cot² x

Given

tanθ = $\sqrt{3}$ , then cotθ = $\frac{1}{\sqrt{3} }$

csc²θ = 1 + ($\frac{1}{\sqrt{3} }$ )² = 1 + $\frac{1}{3}$ = $\frac{4}{3}$

cscθ = ± $\sqrt{\frac{4}{3} }$ = ± $\frac{2}{\sqrt{3} }$

Since θ is in 3rd quadrant, then cscθ < 0

cscθ = - $\frac{2}{\sqrt{3} }$ × $\frac{\sqrt{3} }{\sqrt{3} }$ = - $\frac{2\sqrt{3} }{3}$