Question

Find the value of the following \[\dfrac{d}{dx}\dfrac{2}{\pi }\sin {{x}^{\circ }}=\]
1.\[\left( \dfrac{\pi }{180} \right)\cos x\]
2.\[\left( \dfrac{1}{90} \right)\cos x\]
3.\[\left( \dfrac{\pi }{90} \right)\cos x\]
4.\[\left( \dfrac{2}{90} \right)\cos x\]

Answer 1

Hint:In order to differentiate \[\dfrac{d}{dx}\dfrac{2}{\pi }\sin {{x}^{\circ }}\], firstly we will be converting the given degrees into radians by considering the value to \[{{1}^{\circ }}\]. Then after converting into radians, we will be differentiating the given function with respect to \[x\]. After differentiating with respect to \[x\], upon solving it, we will be obtaining our required answer.

Complete step-by-step solution:
Now let us learn about radians and degrees in trigonometry. Generally, these are considered as the measures to measure the angles. A radian can be defined as the amount an angle should open to capture an arc of a circle's circumference of equal length to the circle’s radius. So one radian is equal to \[180\pi \] degrees which is \[{{53.7}^{\circ }}\]. A degree can be divided into minutes and seconds. We can convert degrees into radians by Angle in radian = Angle in degree\[\times \dfrac{\pi }{180}\].
Now let us start differentiating the function given i.e. \[\dfrac{d}{dx}\dfrac{2}{\pi }\sin {{x}^{\circ }}\].
Firstly let us consider it as \[y=\dfrac{d}{dx}\dfrac{2}{\pi }\sin {{x}^{\circ }}\]
Now let us convert the given function which is in degrees to radians.
We know that \[{{1}^{\circ }}\]\[=\dfrac{\pi }{180}\] radians.
So upon converting, we get
\[y=\dfrac{2}{\pi }\sin \left( \dfrac{\pi }{180} \right)x\]
Now let us differentiate with respect to \[x\], on both sides. We get
\[\begin{align}
  & \Rightarrow y=\dfrac{2}{\pi }\sin \left( \dfrac{\pi }{180} \right)x \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{2}{\pi }\cos \left( \dfrac{\pi }{180} \right)x\times \left( \dfrac{\pi }{180} \right) \\
\end{align}\]
Because we know that \[\dfrac{d}{dx}\sin x=\cos x\]. Here we have used the multiplication rule of differentiation in order to differentiate.
Now upon solving the obtained equation, we get
\[\begin{align}
  & \Rightarrow \dfrac{1}{90}\cos \left( \dfrac{\pi }{180} \right)x \\
 & \Rightarrow \dfrac{1}{90}\cos {{x}^{\circ }} \\
\end{align}\]
As we know that \[{{1}^{\circ }}\]\[=\dfrac{\pi }{180}\] radians.
\[\therefore \] Option 2 is the correct answer.

Note:While differentiating the functions, we must apply the apt rule for solving. If we opt for complicated rules for solving, then the problem would be lengthy and time consuming. We have converted into radians for our easy calculation. The commonly committed error could be not differentiating correctly.