Answer:
cotΘ = - [tex]\sqrt{2}[/tex]
Step-by-step explanation:
Using the trigonometric identities
cot x = [tex]\frac{cosx}{sinx}[/tex]
sin²x + cos²x = 1 ⇒ cosx = ± [tex]\sqrt{1-sin^2x}[/tex]
Since Θ is in second quadrant then cosΘ < 0
cosΘ = - [tex]\sqrt{1-(\frac{\sqrt{3} }{3} }[/tex])^2
= - [tex]\sqrt{1-\frac{1}{3} }[/tex] = - [tex]\sqrt{\frac{2}{3} }[/tex] = - [tex]\frac{\sqrt{2} }{\sqrt{3} }[/tex]
Hence
cotΘ = [tex]\frac{-\frac{\sqrt{2} }{\sqrt{3} } }{\frac{\sqrt{3} }{3} }[/tex]
= - [tex]\frac{\sqrt{2} }{\sqrt{3} }[/tex] × [tex]\frac{3}{\sqrt{3} }[/tex]
= - [tex]\frac{3\sqrt{2} }{3}[/tex] = - [tex]\sqrt{2}[/tex]
