Question

The value of given trigonometric equation $\sin ({{37}^{\circ }})\times \cos ({{53}^{\circ }})=$

Answer 1

Hint: Convert the given equation of trigonometry functions to only one function either of only sine function or of only cos function. Use the below identity for the conversion:

Complete step-by-step solution -
$\begin{align}
  & \sin \left( 90-\theta \right)=\cos \theta \\
 & \cos \left( 90-\theta \right)=\sin \theta \\
\end{align}$
values of sin37 and cos53 are the same and given as $\dfrac{3}{5}$.
Here, we have expression $\sin {{37}^{\circ }}\times \cos {{53}^{\circ }}$
Let the value of this expression be M.
SO, we can write equation as
$M=\sin {{37}^{\circ }}\times \cos {{53}^{\circ }}........(i)$
Let us solve the given expression by converting the whole equation to sin form of cos form.So, can get both by two different cases given below:
Case 1:In first case, we change $\cos \theta $ to $\sin \theta $ with the help of identity $\cos \left( 90-\theta \right)=\sin \theta $ .So, as we need to convert $'\cos {{53}^{\circ }}'$ term to sin function. So, we can write $\cos {{53}^{\circ }}$as $\cos \left( 90-37 \right)$ as well. And now we can use the above identity i.e. $\cos \left( 90-\theta \right)=\sin \theta $ in following way
$\cos \left( 90-37 \right)=\sin 37$
Now, we can get value of M from equation (i)
$\begin{align}
  & M=\sin 37\times \sin 37 \\
 & ={{\left( \sin 37 \right)}^{2}}
\end{align}$
Case2: In this case, we change $\sin \theta $ to $\cos \theta $ with the help of identity $\sin \left( 90-\theta \right)=\cos \theta $.So, we can change $\sin {{37}^{\circ }}$ term in the given expression with the help of above identity in the following way:
Put $\theta ={{53}^{\circ }}$ in the equation $\sin \left( 90-\theta \right)=\cos \theta $
So, we get
$\begin{align}
  & \sin \left( 90-53 \right)=\cos {{53}^{\circ }} \\
 & \sin {{37}^{\circ }}=\cos {{53}^{\circ }} \\
\end{align}$
Now, hence we can get value of M from the equation (i) as
$\begin{align}
  & M=\cos {{53}^{\circ }}\times \cos {{53}^{\circ }} \\
 & M={{\left( \cos {{53}^{\circ }} \right)}^{2}} \\
\end{align}$
Now, we know values of $\sin {{37}^{\circ }}$ and $\cos {{53}^{\circ }}$ can be given as
$\sin {{37}^{\circ }}=\dfrac{3}{5}$ and $\cos {{53}^{\circ }}=\dfrac{3}{5}$
Hence, we can get value of M from both the cases as
$M={{\left( \dfrac{3}{5} \right)}^{2}}=\dfrac{9}{25}$
So, we get value of $\sin {{37}^{\circ }}\times \cos {{53}^{\circ }}$ is $\dfrac{9}{25}$

Note: One may put direct values of $\sin {{37}^{\circ }}$ and $\cos {{53}^{\circ }}$as $\dfrac{3}{5}$ to the given expression as well. We took two approaches for better understanding of the question and the trigonometric identities as well. Values of $\sin {{37}^{\circ }}$ and $\cos {{53}^{\circ }}$cannot be calculated directly. It is directly used in the expression as $\dfrac{3}{5}$ from the table of sin and cos functions.
Value of $\sin {{37}^{\circ }}$ or $\cos {{53}^{\circ }}$ is not exactly $\dfrac{3}{5}=0.6$.Exact value of $\sin {{37}^{\circ }}$ is given as 0.6018150…….So, value of $\sin {{37}^{\circ }}$ and $\cos {{53}^{\circ }}$ used here is approximate value.