Question

If $\sin \theta =\frac{5}{13}$. Write the values of all other trigonometric ratios.

Answer 1

$\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{5}{13}$
In $△ABC$, $\angle B={90}^o$, $\angle C=\theta$
${AC}^2={AB}^2+{BC}^2$ ....... (By Pythagoras theorem)
$\therefore {BC}^2={AC}^2-{AB}^2={13}^2-5^2=169-25=144$
$\therefore BC=\sqrt{144}=12$

$\cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}=\frac{12}{13}$

$\tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{5}{12}$

$\csc \theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}=\frac{13}{5}$

$\sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}=\frac{13}{12}$

$\cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}=\frac{12}{5}$