Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 87

Show that $\displaystyle \frac{d}{d \theta} (\sin \theta) = \frac{\pi}{180} \cos \theta$ by using Chain Rule such that $\theta$ is measured in degrees.


$
\begin{equation}
\begin{aligned}

\frac{d}{d \theta} =& \frac{d}{d \theta} \left( \sin \frac{\pi}{180} \theta \right) \text{ with $\theta$ in radians. So,}
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\\
\frac{d}{d \theta} \left( \sin \frac{\pi}{180} \theta \right) =& \frac{d}{d\left( \frac{\pi \theta}{180} \right) } \left( \sin \frac{\pi}{180} \theta \right)
\\
\\
\frac{d}{d \theta} \left( \sin \frac{\pi}{180} \theta \right) =& \cos \frac{\pi \theta}{180} \cdot \frac{d}{d \theta} \left( \frac{\pi \theta}{180} \right)
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\\
\frac{d}{d \theta} \left( \sin \frac{\pi}{180} \theta \right) =& \cos \frac{\pi \theta}{180} \cdot \frac{\pi}{180} = \frac{\pi}{180} \cdot \cos \left( \frac{\pi \theta}{180} \right)
\\
\\

\end{aligned}
\end{equation}
$




But we have in degrees,

$\displaystyle \frac{d}{d \theta} \left( \sin \frac{\pi}{180} \theta \right) = \frac{\pi}{180} \cdot \cos \theta$