Question

If cot θ = 2, find the value of all T-ratios of θ.

Answer 1

Let us first draw a right $∆$ABC, right angled at B and $\angle C=\theta$.
Now, we know that cot $\theta$ = $\frac{\mathrm{base}}{\mathrm{Perpendicular}}$ = $\frac{BC}{AB}$ = 2.


So, if BC = 2k, then AB = k, where k is a positive number.
Now, using Pythagoras theorem, we have:
AC² = AB² + BC² = (2k)² + (k)²
⇒ AC² = 4k² + k² = 5k²
⇒ AC = $\sqrt{5}$k
Now, finding the other T-ratios using their definitions, we get:
sin $\theta$ = $\frac{AB}{AC}$ = $\frac{k}{\sqrt{5}k}=\frac{1}{\sqrt{5}}$
cos $\theta$ = $\frac{BC}{AC}=\frac{2k}{\sqrt{5}k}=\frac{2}{\sqrt{5}}$

∴ tan $\theta$ = $\frac{1}{cot\theta }=\frac{1}{2}$, cosec $\theta$ = $\frac{1}{\sin \theta }=\sqrt{5}$ and sec $\theta$ = $\frac{1}{\cos \theta }=\frac{\sqrt{5}}{2}$