Question

Express the following angles in radians: $\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{20}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{}\mathbf{50}\mathbf{.}{\mathbf{0}}^{\mathbf{°}\mathbf{}}\mathbf{,}{\mathbf{100}}^{\mathbf{°}}$Convert the following angles to degrees:$\mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{330}\mathbf{}\mathbf{rad}\mathbf{}\mathbf{,}\mathbf{}\mathbf{\left(}\mathbf{e}\mathbf{\right)}\mathbf{2}\mathbf{.}\mathbf{10}\mathbf{}\mathbf{rad}\mathbf{}\mathbf{,}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{}\mathbf{7}\mathbf{.}\mathbf{70}\mathbf{}\mathbf{rad}$

Short answer

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathrm{Angles}\mathrm{in}\mathrm{radians};20.0^{°}=0.349\mathrm{rad} \\ 50.0^{°}=0.873\mathrm{rad} \\ 100.0^{°}=1.75\mathrm{rad} \\ \mathbf{b}\mathbf{)}\mathrm{Angles}\mathrm{in}\mathrm{degrees}:0.330\mathrm{rad}=18.9^{°} \\ 0.330\mathrm{rad}={120}^{°} \\ 0.330\mathrm{rad}={441}^{°} \end{gathered}$$

Step-by-step solution

Understanding the relation between degree and radian.

${\mathbf{360}}^{\mathbf{°}}\mathbf{}\mathbf{of}\mathbf{}\mathbf{2}\mathbf{\pi }\mathbf{}\mathbf{rad}$of makes up a complete circle angle. This means a circle subtends an angle with radian $\mathbf{2}\mathbf{\pi }$and degree 360° at its centre. An angle's measure can be converted from one unit to another based on the following formula.

$$\begin{gathered} {360}^{°}=2\mathrm{\pi }\mathrm{rad}\left(1\right) \\ \mathrm{Given}\mathrm{are}\mathrm{the}\mathrm{angles}\mathrm{in}\mathrm{degrees}{20}^{°},{50}^{°},{100}^{°}\mathrm{and}\mathrm{Angles}\mathrm{in}\mathrm{radians} \\ 0.330\mathrm{rad}.2.10\mathrm{rad},7.70\mathrm{rad}. \end{gathered}$$

To calculate angle in radians part a)

Using equation (i) the angle can be expressed in radians as

$$\begin{gathered} 20.0^{°}=\left(\right(20.0^{°})\frac{2\mathrm{\pi }}{{360}^{°}}\mathrm{rad}=0.349\mathrm{rad} \\ 50.0^{°}=((50.0^{°})\frac{2\mathrm{\pi }}{{360}^{°}}\mathrm{rad}=0.873\mathrm{rad} \\ 100.0^{°}=\left(\right(100.0^{°})\frac{2\mathrm{\pi }}{{360}^{°}}\mathrm{rad}=1.75\mathrm{rad} \end{gathered}$$

To calculate angle in degrees part b)

Using equation (i) the angle can be expressed in degrees as

$$\begin{gathered} 0.330\mathrm{rad}=(0.330\mathrm{rad})\frac{{360}^{°}}{2\mathrm{\pi }\mathrm{rad}}=18.9^{°} \\ 2.10\mathrm{rad}=(2.10\mathrm{rad})\frac{{360}^{°}}{2\mathrm{\pi }\mathrm{rad}}={120}^{°} \\ 7.70\mathrm{rad}=(7.70\mathrm{rad})\frac{{360}^{°}}{2\mathrm{\pi }\mathrm{rad}}={441}^{°} \end{gathered}$$