Question

If $15\cot A=8$, find the values of sinA and secA.

Answer 1

We have,

$$\begin{gathered} 15\cot A=8 \\ \Rightarrow \cot A=\frac{8}{15} \end{gathered}$$

As,

$$\begin{gathered} {\mathrm{cosec}}^{2}A=1+{\cot }^{2}A \\ =1+{\left(\frac{8}{15}\right)}^2 \\ =1+\frac{64}{225} \\ =\frac{225+64}{225} \end{gathered}$$
$$\begin{gathered} \Rightarrow {\mathrm{cosec}}^{2}A=\frac{289}{225} \\ \Rightarrow \mathrm{cosec}A=\sqrt{\frac{289}{225}} \\ \Rightarrow \mathrm{cosec}A=\frac{17}{15} \\ \Rightarrow \frac{1}{\sin A}=\frac{17}{15} \\ \Rightarrow \sin A=\frac{15}{17} \end{gathered}$$

Also,

$$\begin{gathered} {\cos }^{2}A=1-{\sin }^{2}A \\ =1-{\left(\frac{15}{17}\right)}^2 \\ =1-\frac{225}{289} \\ =\frac{289-225}{289} \end{gathered}$$
$$\begin{gathered} \Rightarrow {\cos }^{2}A=\frac{64}{289} \\ \Rightarrow \cos A=\sqrt{\frac{64}{289}} \\ \Rightarrow \cos A=\frac{8}{17} \\ \Rightarrow \frac{1}{\sec A}=\frac{8}{17} \\ \Rightarrow \sec A=\frac{17}{8} \end{gathered}$$