Question

If tan $\mathrm{\theta }=\frac{15}{8}$ find the values 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 tan $\theta$ = $\frac{\mathrm{Perpendicular}}{\mathrm{Base}}$ = $\frac{AB}{BC}$ = $\frac{15}{8}$.

So, if BC = 8k, then AB = 15k, where k is a positive number.
Now, using Pythagoras theorem, we have:
AC² = AB² + BC² = (15k)² + (8k)²
⇒ AC² = 225k² + 64k² = 289k²
⇒ AC = 17k

Now, finding the other T-ratios using their definitions, we get:
sin $\theta$ = $\frac{AB}{AC}$ = $\frac{15k}{17k}=\frac{15}{17}$
cos $\theta$ = $\frac{BC}{AC}=\frac{8k}{17k}=\frac{8}{17}$

∴ cot $\theta$ = $\frac{1}{\tan \theta }=\frac{8}{15}$, cosec $\theta$ = $\frac{1}{\sin \theta }=\frac{17}{15}$ and sec $\theta$ = $\frac{1}{\cos \theta }=\frac{17}{8}$