Question

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.
$cosec\theta =\sqrt{10}$

Answer 1

Given,

$\csc \theta =\sqrt{10}$

we know that

$cosec\theta =\frac{Hypotenuse}{Oppositeside}$

Consider the figure above,

From Pythagoras theorem,

$A{C}^2=A{B}^2+B{C}^2$

$(\sqrt{10}{)}^2=1^2+B{C}^2$

$B{C}^2=10-1=9$

$BC=3$

$\sin \theta =\frac{OppositeSide}{Hypotenuse}=\frac{AB}{AC}=\frac{1}{\sqrt{10}}$

$\cos \theta =\frac{adjacentSide}{Hypotenuse}=\frac{BC}{AC}=\frac{3}{\sqrt{10}}$

$\tan \theta =\frac{OppositeSide}{AdjacentSide}=\frac{1}{3}$

$\sec \theta =\frac{1}{\cos \theta }=\frac{1}{\frac{3}{\sqrt{10}}}=\frac{\sqrt{10}}{3}$

$\cot \theta =\frac{1}{\tan \theta }=\frac{1}{\frac{1}{3}}=3$