Question

Match the following if in $\Delta$ABC, $\angle B={90}^0,$ $\angle C=\theta$ and AB = 9 cm and BC = 12 cm.

• A. $cosec\theta$
• B. $cot\theta$
• C. $sec\theta$

Answer 1

Given,
In $\Delta$ABC, $\angle B={90}^0,$ $\angle C=\theta$
AB = 9 cm and BC = 12 cm

By pythagoras theorem,
$(AC{)}^2=(AB{)}^2+(BC{)}^2$
$(AC{)}^2=(9{)}^2+(12{)}^2$
$(AC{)}^2=81+144$
$(AC{)}^2=225$
$AC=15cm$

Case 1:

$sin\theta$ = $\frac{AB}{AC}=\frac{9}{15}$
$cosec\theta$ = $\frac{1}{sin\theta }$
$\therefore$ $cosec\theta$ = $\frac{1}{\frac{9}{15}}=\frac{15}{9}$

Case 2:

$tan\theta$ = $\frac{AB}{BC}=\frac{9}{12}$
$cot\theta$ = $\frac{1}{tan\theta }$
$\therefore$ $cot\theta$ = $\frac{1}{\frac{9}{12}}=\frac{12}{9}$

Case 3:

$cos\theta$ = $\frac{BC}{AC}=\frac{12}{15}$
$sec\theta$ = $\frac{1}{cos\theta }$
$\therefore$ $sec\theta$ = $\frac{1}{\frac{12}{15}}=\frac{15}{12}$