Answer:
The exact value of cotФ is [tex]\frac{\sqrt{7}}{3}[/tex]
Step-by-step explanation:
Given as:
The value of cosec Ф = [tex]\frac{-4}{3}[/tex]
Let the value of cotФ = x
Now, According to question
∵ sinФ = [tex]\frac{1}{cosec\Theta }[/tex] .....1
Put the value of cosec Ф = [tex]\frac{-4}{3}[/tex] in eq 1
i.e sinФ = [tex]\frac{1}{\frac{-4}{3} }[/tex]
Or, sinФ = [tex]\frac{-3}{4}[/tex]
Again
∵ cosФ = [tex]\sqrt{1-sin^{2}\Theta }[/tex]
So, cosФ = [tex]\sqrt{1-(\frac{-3}{4})^{2}}[/tex]
Or, cosФ = [tex]\sqrt{1-(\frac{9}{16})}[/tex]
Or, cosФ = [tex]\sqrt{\frac{16 - 9}{16})}[/tex]
∴ cosФ = [tex]\frac{\sqrt{7}}{4}[/tex]
Again
we know that cotФ = [tex]\frac{cos\Theta }{sin\Theta }[/tex]
So, cotФ = [tex]\frac{\frac{\sqrt{7}}{4}}{\frac{-3}{4}}[/tex]
Or, cotФ = [tex]\frac{-\sqrt{7}}{3}[/tex]
As according to question sinФ lies in third quadrant
So, cotФ = [tex]\frac{\sqrt{7}}{3}[/tex]
Hence, The exact value of cotФ is [tex]\frac{\sqrt{7}}{3}[/tex] . Answer
