Find the values of other five trigonometric functions if cos⁡(x)=-12,x lies in the third quadrant.

sin ⁡ x = - 3 2 , tan ⁡ x = 3 , cot ⁡ x = 1 3 , sec ⁡ x = - 2 , cosec ⁡ x = - 2 3

Find the values of other five trigonometric functions if cos⁡(x)=-12,x lies in the third quadrant.

sin ⁡ x = - 3 2 , tan ⁡ x = 3 , cot ⁡ x = 1 3 , sec ⁡ x = - 2 , cosec ⁡ x = - 2 3

sin x = 3 2 , tan x = - 1 3 , cot x = 2 3 , sec x = - 12 , cosec x = - 12 3

sin x = - 3 4 , tan x = 3 3 , cot x = - 3 , sec x = - 10 , cosec x = 2 3

sin x = 2 2 , tan x = 1 2 , cot x = 5 , sec x = - 8 , cosec x = - 2 5

sin ⁡ x = - 3 2 , tan ⁡ x = 3 , cot ⁡ x = 1 3 , sec ⁡ x = - 2 , cosec ⁡ x = - 2 3

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Find the values of other five trigonometric functions if $\cos (x) = \frac{- 1}{2}, x$ lies in the third quadrant.

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\sin (x) = \frac{\sqrt{3}}{2}, \tan (x) = \frac{- 1}{\sqrt{3}}, \cot (x) = 2 \sqrt{3}, \sec (x) = - 12, cosec (x) = \frac{- 12}{\sqrt{3}}

\sin (x) = \frac{- \sqrt{3}}{4}, \tan (x) = \frac{3}{\sqrt{3}}, \cot (x) = - \sqrt{3}, \sec (x) = - 10, cosec (x) = \frac{2}{\sqrt{3}}

\sin (x) = \frac{\sqrt{2}}{2}, \tan (x) = \frac{1}{\sqrt{2}}, \cot (x) = \sqrt{5}, \sec (x) = - 8, cosec (x) = \frac{- 2}{\sqrt{5}}

\sin x = - \frac{\sqrt{3}}{2}

\tan x = \sqrt{3}

\cot x = \frac{1}{\sqrt{3}}

\sec x = - 2

cosec x = - \frac{2}{\sqrt{3}}

answer is D.

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Detailed Solution

Given,

\cos (x) = \frac{- 1}{2}

Now by using trigonometry identity

\sin x = \sqrt{1 - \cos^{2} x}

\sin x = \sqrt{1 - \frac{1}{4}}

\sin x = \sqrt{\frac{3}{4}}

\sin x = - \frac{\sqrt{3}}{2}

Also,

\tan x = \frac{\sin x}{\cos x}

\tan x = \sqrt{3}

Now, we know that,

\cot x = \frac{1}{\tan x}

\cot x = \frac{1}{\sqrt{3}}

The value of sec x,

\sec x = \frac{1}{\cos x}

\sec x = - 2

Similarly the value of cosec x,

cosec x = \frac{1}{\sin x}

cosec x = - \frac{2}{\sqrt{3}}

Hence the five-trigonometry function are

\sin x = - \frac{\sqrt{3}}{2}

\tan x = \sqrt{3}

\cot x = \frac{1}{\sqrt{3}}

\sec x = - 2

cosec x = - \frac{2}{\sqrt{3}}

Option 4 is correct.

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