Question

If $\sin \left(\mathrm{A}\right)+\sin \left(\mathrm{B}\right)+\sin \left(\mathrm{C}\right)+s\mathrm{in}\left(\mathrm{D}\right)=4$ then find the value of $\sin \left(\mathrm{A}\right)\times \sin \left(\mathrm{B}\right)\times \sin \left(\mathrm{C}\right)\times \sin \left(\mathrm{D}\right)$.

Answer 1

Step 1: Compare the range of LHS and RHS.

Range of $\sin \left(\mathrm{x}\right)\forall \mathrm{x}\in \mathrm{ℝ}$ is $\left[-1,1\right]$

$$\begin{gathered} \Rightarrow -1\le \sin \left(\mathrm{A}\right)\le 1 \\ \Rightarrow -1\le \sin \left(\mathrm{B}\right)\le 1 \\ \Rightarrow -1\le \sin \left(\mathrm{C}\right)\le 1 \\ \Rightarrow -1\le \sin \left(\mathrm{D}\right)\le 1 \end{gathered}$$

Adding all four equations

$\Rightarrow -4\le \sin \left(\mathrm{A}\right)+\sin \left(\mathrm{B}\right)+\sin \left(\mathrm{C}\right)+\sin \left(\mathrm{D}\right)\le 4$

For $\sin \left(\mathrm{A}\right)+\sin \left(\mathrm{B}\right)+\sin \left(\mathrm{C}\right)+\sin \left(\mathrm{D}\right)$to be equal to $4$, $\sin \left(\mathrm{A}\right)=\sin \left(\mathrm{B}\right)=\sin \left(\mathrm{C}\right)=\sin \left(\mathrm{D}\right)=1$

Step 2: Calculate the given expression.

Since $\sin \left(\mathrm{A}\right)=\sin \left(\mathrm{B}\right)=\sin \left(\mathrm{C}\right)=\sin \left(\mathrm{D}\right)=1$

$\Rightarrow \sin \left(\mathrm{A}\right)\times \sin \left(\mathrm{B}\right)\times \sin \left(\mathrm{C}\right)\times \sin \left(\mathrm{D}\right)=1^4=1$

Hence, value of $\sin \left(\mathrm{A}\right)\times \sin \left(\mathrm{B}\right)\times \sin \left(\mathrm{C}\right)\times \sin \left(\mathrm{D}\right)$ is $1$.