Question

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

Answer 1

$sin\theta =\frac{\sqrt{3}}{2}$

We know that,

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

Consider the figure above,

From Pythagoras theorem,

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

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

$A{B}^2=4-3=1$

$AB=1$

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

$\cos \theta =\frac{AdjacentSide}{Hypotenuse}=\frac{1}{2}$

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

$\sec \theta =\frac{1}{\cos \theta }=\frac{1}{\frac{1}{2}}=2$

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