Question

In a triangle $ABC$ right angled at $B$, if $\tan \left(A\right)=\frac{1}{\sqrt{3}}$ find the value of $\sin \left(\mathrm{A}\right)\cos \left(\mathrm{C}\right)+\sin \left(\mathrm{C}\right)\cos \left(\mathrm{A}\right)$.

Answer 1

Step 1: Find the value of angles of the given triangle

In the question, it is given that a triangle $ABC$ is right angled at $B$, if $\tan \left(A\right)=\frac{1}{\sqrt{3}}$.

$\begin{array}{rcl}\angle \mathrm{B}& =& 90°\left[\because \mathrm{given}\mathrm{ABC}\mathrm{is}\mathrm{right}\mathrm{angled}\mathrm{at}\mathrm{B}\right]\\ & \Rightarrow & \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180°\left[\because \mathrm{Angle}\mathrm{Sum}\mathrm{Property}\right]\\ & \Rightarrow & \angle \mathrm{A}+\angle \mathrm{C}=90°\end{array}$

Step 2: Find the required value

We know that, $\sin \left(\mathrm{A}+\mathrm{C}\right)=\sin \left(\mathrm{A}\right)\cos \left(\mathrm{C}\right)+\sin \left(\mathrm{C}\right)\cos \left(\mathrm{A}\right)$.
So, the value of the given expression can be calculated as follows:

$$\begin{gathered} \sin \left(\mathrm{A}\right)\cos \left(\mathrm{C}\right)+\sin \left(\mathrm{C}\right)\cos \left(\mathrm{A}\right)=\sin \left(90°\right)\left[\because \mathrm{A}+\mathrm{C}=90°\right] \\ \Rightarrow \sin \left(\mathrm{A}\right)\cos \left(\mathrm{C}\right)+\sin \left(\mathrm{C}\right)\cos \left(\mathrm{A}\right)=1\left[\because \sin \left(90°\right)=1\right] \end{gathered}$$

Hence the value of $\sin \left(\mathrm{A}\right)\cos \left(\mathrm{C}\right)+\sin \left(\mathrm{C}\right)\cos \left(\mathrm{A}\right)$ is $1$.