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Q.

If the value of $\tan α = \frac{5}{12}$ , find all other trigonometric ratios.

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a

\sin (x) = \frac{5}{13}

,

\cos (x) = \frac{12}{13}, cosec (x) = \frac{13}{5}, \sec (x) = \frac{13}{12}

,

\cot (x) = \frac{12}{5}

b

\sin (x) = 1

,

\cos (x) = 2, cosec (x) = \frac{13}{5}, \sec (x) = \frac{13}{12}

,

\cot (x) = \frac{12}{5}

c

\sin (x) = 2

,

\cos (x) = \frac{12}{13}, cosec (x) = \frac{13}{5}, \sec (x) = \frac{13}{12}

,

\cot (x) = 4

d

\sin (x) = \frac{5}{13}

,

\cos (x) = \frac{2}{13}, cosec (x) = \frac{1}{5}, \sec (x) = 6

,

\cot (x) = \frac{12}{5}

answer is A.

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Detailed Solution

Given,

\tan (x) = \frac{5}{12}

Suppose there is a right-angled triangle ABC, right angle at B.
Question Image

Question Image

Now, we know that,

\tan θ = \frac{Perpendicular side}{Base side}

On comparing with given value we get,
Perpendicular side = 5
Base side = 12
Thus, hypotenuse side from Pythagoras theorem can be given as,

{(AC)}^{2} = 5^{2} + 12^{2}

{(AC)}^{2} = 25 + 144

{(AC)}^{2} = 169

AC = \sqrt{{(13)}^{2}}

AC = 13

Now from trigonometry properties,

si n (x) = \frac{Perpendicular side}{hypotenuse side} = \frac{5}{13}

co s (x) = \frac{Base side}{hypotenuse side} = \frac{12}{13}

And,

cosec (x) = \frac{1}{\sin x} = \frac{13}{5}

\sec (x) = \frac{1}{\cos (x)} = \frac{13}{12}

\cot (x) = \frac{1}{\tan (x)} = \frac{12}{5}

So, we get,

\sin (x) = \frac{5}{13}

,

\cos (x) = \frac{12}{13}, cosec (x) = \frac{13}{5}, \sec (x) = \frac{13}{12}

,

\cot (x) = \frac{12}{5}

Correct option is 1.

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