Question

If cos $\mathrm{\theta }=\frac{7}{25}$ 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 cos $\theta$ = $\frac{\mathrm{Base}}{\mathrm{hypotenuse}}$= $\frac{BC}{AC}$ = $\frac{7}{25}$.

So, if BC = 7k, then AC = 25k, where k is a positive number.
Now, using Pythagoras theorem, we have:
AC² = AB² + BC²
⇒ AB² = AC² $-$ BC² = (25k)² $-$ (7k)².
⇒ AB² = 625k² $-$ 49k² = 576k²
⇒ AB = 24k
Now, finding the other trigonometric ratios using their definitions, we get:
sin $\theta$ = $\frac{AB}{AC}$ = $\frac{24k}{25k}=\frac{24}{25}$
tan $\theta$ = $\frac{AB}{BC}=\frac{24k}{7k}=\frac{24}{7}$
∴ cot $\theta$ = $\frac{1}{\tan \theta }=\frac{7}{24}$, cosec $\theta$ = $\frac{1}{\sin \theta }=\frac{25}{24}$ and sec $\theta$ = $\frac{1}{\cos \theta }=\frac{25}{7}$