In each of the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.$ \tan \alpha=\frac{5}{12} $

Given:

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

To do:

We have to find the values of the other trigonometric ratios. 

Solution:  

We know that,

In a right-angled triangle $ABC$ with right angle at $B$,

By Pythagoras theorem,

$AC^2=AB^2+BC^2$

By trigonometric ratios definitions,

$sin\ A=\frac{Opposite}{Hypotenuse}=\frac{BC}{AC}$

$cos\ A=\frac{Adjacent}{Hypotenuse}=\frac{AB}{AC}$

$tan\ A=\frac{Opposite}{Adjacent}=\frac{BC}{AB}$

$cosec\ A=\frac{Hypotenuse}{Opposite}=\frac{AC}{BC}$

$sec\ A=\frac{Hypotenuse}{Adjacent}=\frac{AC}{AB}$

$cot\ A=\frac{Adjacent}{Opposite}=\frac{AB}{BC}$

Here,

Let $\tan \alpha=\frac{BC}{AB}=\frac{5}{12}$

$AC^2=AB^2+BC^2$

$\Rightarrow AC^2=(12)^2+(5)^2$

$\Rightarrow AC^2=144+25$

$\Rightarrow AC=\sqrt{169}=13$

Therefore,

$sin\ \alpha=\frac{BC}{AC}=\frac{5}{13}$

$cos\ \alpha=\frac{AB}{AC}=\frac{12}{13}$

$cosec\ \alpha=\frac{AC}{BC}=\frac{13}{5}$

$sec\ \alpha=\frac{AC}{AB}=\frac{13}{12}$

$cot\ \alpha=\frac{AB}{BC}=\frac{12}{5}$

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Updated on: 10-Oct-2022

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