Question

Given $15\cot A=8$, find $\sin A$ and $\cos A$

Answer 1

Given: $15\cot A=8$

$\Rightarrow \cot A=\frac{8}{15}$

$\Rightarrow {\cot }^{2}A=\frac{64}{225}$

$\Rightarrow {\text{cosec}}^{2}A-1=\frac{64}{225}$, [Since${\cot }^{2}A={\text{cosec}}^{2}A-1$]

$\Rightarrow {\text{cosec}}^{2}A=1+\frac{64}{225}$

$\Rightarrow {\text{cosec}}^{2}A=\frac{225+64}{225}$

$\Rightarrow {\text{cosec}}^{2}A=\frac{289}{225}$

$\Rightarrow \text{cosec}A=\frac{17}{15}$

$\Rightarrow \sin A=\frac{15}{17}$, ---(1) [Since, $\text{cosec}A=\frac{1}{\sin A}$]

$\Rightarrow {\sin }^{2}A=\frac{225}{289}$,

$\Rightarrow 1-{\cos }^{2}A=\frac{225}{289}$, [Since, ${\sin }^{2}A=1-{\cos }^{2}A$]

$\Rightarrow {\cos }^{2}A=1-\frac{225}{289}$

$\Rightarrow {\cos }^{2}A=\frac{289-225}{289}$

$\Rightarrow {\cos }^{2}A=\frac{64}{289}$

$\Rightarrow \cos A=\frac{8}{17}$-------(2)