Question

Match the length of third side for each of the following cases:

• A. 1
• B. cos $\theta$
• C. cot $\theta$

Answer 1

For (i) case,
$AB$ = 1 and $AC$ = $cosec$ $\theta$
Applying pythagoras theorem,
$A{C}^2=A{B}^2+B{C}^2$
$cose{c}^2\theta =1^2+B{C}^2$
$\Rightarrow$ $BC=\sqrt{cose{c}^2\theta -1}=\sqrt{co{t}^2\theta }$
$\therefore BC=cot\theta$ [from Identity]

For (ii) case,
$AB$ = $\frac{1}{cos\theta }$ = $sec\theta$ and $BC$ = $tan\theta$
Applying pythagoras theorem,
$A{C}^2=A{B}^2+B{C}^2$
$A{C}^2=se{c}^2\theta +ta{n}^2\theta$
$\Rightarrow$ $AC=\sqrt{se{c}^2\theta +ta{n}^2\theta }=\sqrt{1}$
$\therefore AC=1$ [from Identity]

For (iii) case,
$AB$ = $\frac{1}{cosec\theta }$ = $sin\theta$ and $AC$ = $1$
Applying pythagoras theorem,
$A{C}^2=A{B}^2+B{C}^2$
$1^2=si{n}^2\theta +B{C}^2$
$\Rightarrow$ $BC=\sqrt{1-si{n}^2\theta }=\sqrt{co{s}^2\theta }$
$\therefore BC=cos\theta$ [from Identity]