Find tan θ if sec θ = square root of thirty seven divided by six and sin θ < 0.

Solution :

Given that sin θ < 0,

⇒ sin θ is negative

Then θ can be angle from the third quadrant or fourth quadrant

Case 1: If θ lies in third quadrant

sec θ = - √37/6

tan θ = √(sec²θ -1) = √{(-√37/6)² - 1} = √31/6

Case 2: If θ lies in fourth quadrant

sec θ = √37/6

tan θ = - √(sec²θ -1) = -√{(√37/6)² - 1} = - √31/6

Find tan θ if sec θ = square root of thirty seven divided by six and sin θ < 0

Summary:

The value of tanθ if θ lies in 3rd quadrant = √31/6, The value of tanθ if θ lies in 4th quadrant = -√31/6