Question

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.
$\sin A$ = $\frac{2}{3}$

Answer 1

Given,

$sin$ $A=\frac{2}{3}$

We know that,

$\sin \theta =\frac{OppositeSide}{Hypotenuse}$

Consider the figure above,

From Pythagoras theorem,

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

$3^2=A{B}^2+2^2$

$A{B}^2=9-4=5$

$AB=\sqrt{5}$

$\cos \theta =\frac{AdjacentSide}{Hypotenuse}=\frac{\sqrt{5}}{3}$

$\sec \theta =\frac{1}{\cos \theta }=\frac{1}{\frac{\sqrt{5}}{3}}=\frac{3}{\sqrt{5}}$

$\csc \theta =\frac{1}{\sin \theta }=\frac{1}{\frac{2}{3}}=\frac{3}{2}$

$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}}=\frac{2}{\sqrt{5}}$

$\cot \theta =\frac{1}{\tan \theta }=\frac{\sqrt{5}}{2}$