Question

If $\cos B=$ $\frac{1}{3}$, find the other five trigonometric ratios

Answer 1

We have,

$\cos B=$ $\frac{Base}{Hypotenuse}=\frac{1}{3}$

So, we draw a triangle ABC, right angled at C such that

Base $=BC=1$ unit and, Hypotenuse $=AB=3$ units.

By pythagoras theorem, we have

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

$\Rightarrow 3^2=1^2+A{C}^2$

$\Rightarrow A{C}^2=9-1=8$

$\Rightarrow AC=\sqrt{8}=2\sqrt{2}$

When we consider the t-ratios of $\angle B,$ we have

Base $=BC=1$, Perpendicular $=AC=$ $2\sqrt{2}$, and Hypotenuse $=AB=3$

$\therefore sinB=\frac{Perpendicular}{Hypotenuse}=\frac{2\sqrt{2}}{3}$

$tanB=\frac{Perpendicular}{Base}=\frac{2\sqrt{2}}{1}=2\sqrt{2}$

$cosecB=\frac{Hypotenuse}{Perpendicular}=\frac{3}{2\sqrt{2}}$

$secB=\frac{Hypotenuse}{Base}=\frac{3}{1}=3$

$cotB=\frac{Base}{Perpendicular}=\frac{1}{2\sqrt{2}}$