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Q.

How do you find the value of $\cot (\frac{2 π}{3})$ ?

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a

- \frac{\sqrt{3}}{3}

b

\frac{1}{2}

c

- \frac{1}{2}

d

\frac{\sqrt{3}}{2}

answer is A.

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Detailed Solution

\Rightarrow \frac{2 π}{3} = 120 °

(Convert into degrees)

\Rightarrow 120 ° = 60 °

(Supplementary angle)

\Rightarrow tanθ = \frac{1}{cotθ}

(

\cot

is inverse of

\tan

)

\Rightarrow tanθ = \frac{sinθ}{cosθ}

\Rightarrow \sin (180 ° - x) = x; \cos (180 ° - x) = - x

Finding the value of

\sin

\Rightarrow \sin 120 ° = \sin 60 °

\Rightarrow \frac{\sqrt{3}}{2}

Similarly for

\cos

\Rightarrow \cos 120 ° = - \cos 60 °

\Rightarrow \frac{- 1}{2}

Putting value of

\cos

and

\sin

in

\cot

cotθ = \frac{cosθ}{sinθ}

Putting the value of

\cos

and

\sin

which we found above,

\Rightarrow \frac{\frac{- 1}{2}}{\frac{\sqrt{3}}{2}}

\Rightarrow \frac{- 1}{2} \cdot \frac{2}{\sqrt{3}}

\Rightarrow \frac{- 1}{\sqrt{3}}

\Rightarrow \frac{- 1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}

(Rationalising the denominator)

\Rightarrow \frac{- \sqrt{3}}{3}

Hence, the value of

\cot (\frac{2 π}{3})

is

\frac{- 1}{\sqrt{3}}

or

\frac{- \sqrt{3}}{\sqrt{3}}

.

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