Question

Given: cos A = ${\displaystyle \frac{5}{13}}$

Evaluate:

1. ${\displaystyle \frac{\sin \text{A}–\cot \text{A}}{2\tan \text{A}}}$
2. ${\displaystyle \cot \text{A}+\frac{1}{\cos \text{A}}}$

Answer 1

Consider the diagram below:

cos A = ${\displaystyle \frac{5}{13}}$

i.e. ${\displaystyle \frac{\text{base}}{\text{hypotenuse}}=\frac{5}{13}}$

⇒ ${\displaystyle \frac{\text{AB}}{\text{AC}}=\frac{5}{13}}$

Therefore, if length of AB = 5x, length of AC = 13x

Since

AB² + BC² = AC²          ...[Using Pythagoras Theroem]

(5x)² + BC² = (13x)²

BC² = 169x² – 25x²

BC² = 144x²

∴ BC = 12x                 ...(perpendicular)

Now

tan A = ${\displaystyle \frac{\text{perpendicular}}{\text{base}}=\frac{12x}{5x}=\frac{12}{5}}$

sin A =  ${\displaystyle \frac{\text{perpendicular}}{\text{hypotenuse}} =\frac{12x}{13x}=\frac{12}{13}}$

cot A = ${\displaystyle \frac{\text{base}}{\text{perpendicular}}=\frac{5x}{12x}=\frac{5}{12}}$

(i) ${\displaystyle \frac{\sin \text{A} – \cot \text{A}}{2\tan \text{A}}}$

= ${\displaystyle \frac{\frac{12}{13}–\frac{5}{12}}{2\left(\frac{12}{5}\right)}}$

= ${\displaystyle \frac{79}{156}.\frac{5}{24}}$

= ${\displaystyle \frac{395}{3744}}$

(ii) cot A + ${\displaystyle \frac{1}{\cos \text{A}}}$

= ${\displaystyle \frac{5}{12}+\frac{1}{\frac{5}{13}}}$

= ${\displaystyle \frac{5}{12}+\frac{13}{5}}$

= ${\displaystyle \frac{181}{60}}$