Question

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

Answer 1

3rd option

Explanation:

Using the identities

cot x =
`(1)/(tanx)`

csc² x = 1 + cot² x

Given

tanθ =
`√(3)` , then cotθ =
`(1)/(√(3) )`

csc²θ = 1 + (
`(1)/(√(3) )` )² = 1 +
`(1)/(3)` =
`(4)/(3)`

cscθ = ±
`\sqrt{(4)/(3) }` = ±
`(2)/(√(3) )`

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

cscθ = -
`(2)/(√(3) )` ×
`(√(3) )/(√(3) )` = -
`(2√(3) )/(3)`