Question

Find the degree measure corresponding to the following radian measures:
(i) $\frac{9\mathrm{\pi }}{5}$
(ii) $-\frac{5\mathrm{\pi }}{6}$
(iii) $\left(\frac{18\mathrm{\pi }}{5}\right)$
(iv) (−3)^c
(v) 11^c
(vi) 1^c

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \mathrm{\pi }\mathrm{rad}=180° \\ \therefore 1\mathrm{rad}={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\right)}^{°} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{i}\right)\frac{18\mathrm{\pi }}{5}={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times {\displaystyle \frac{9\mathrm{\pi }}{5}}\right)}^{°} \\ ={\left(36\times 9\right)}^{°} \\ ={324}^{°} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{ii}\right)-\frac{5\mathrm{\pi }}{6}={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times \left(-{\displaystyle \frac{5\mathrm{\pi }}{6}}\right)\right)}^{°} \\ =-{\left(30\times 5\right)}^{°} \\ =-{\left(150\right)}^{°} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{iii}\right){\left(\frac{18\mathrm{\pi }}{5}\right)}^c={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times {\displaystyle \frac{18\mathrm{\pi }}{5}}\right)}^{°} \\ ={\left(36\times 18\right)}^{°} \\ ={648}^{°} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{iv}\right){\left(-3\right)}^c={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times -3\right)}^{°} \\ ={\left(\frac{180}{22}\times 7\times -3\right)}^{°} \\ ={\left(\frac{-3780}{22}\right)}^{°} \\ ={\left(-171{\displaystyle \frac{18}{22}}\right)}^{°} \\ =\left\{-171°{\left({\displaystyle \frac{18}{22}}\times 60\right)}^{\text{'}}\right\} \\ =\left\{-171°{\left(49{\displaystyle \frac{1}{11}}\right)}^{\text{'}}\right\} \\ =-\left\{171°49\text{'}{\left({\displaystyle \frac{1}{11}}\times 60\right)}^{\text{'}\text{'}}\right\} \\ =-\left(171°49\text{'}5.{45}^{\text{'}\text{'}}\right) \\ \approx -\left(171°49\text{'}{5}^{\text{'}\text{'}}\right) \end{gathered}$$


$$\begin{gathered} \left(\mathrm{v}\right){\left(11\right)}^c={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times 11\right)}^{°} \\ ={\left(\frac{180}{22}\times 7\times 11\right)}^{°} \\ ={630}^{°} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{vi}\right){\left(1\right)}^c={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times 1\right)}^{°} \\ ={\left(\frac{180}{22}\times 7\times 1\right)}^{°} \\ ={\left(\frac{630}{11}\right)}^{°} \\ ={\left(57{\displaystyle \frac{3}{11}}\right)}^{°} \\ =57°{\left({\displaystyle \frac{3}{11}}\times 60\right)}^{\text{'}} \\ =57°{\left(16{\displaystyle \frac{4}{11}}\right)}^{\text{'}} \\ =57°16\text{'}{\left({\displaystyle \frac{4}{11}}\times 60\right)}^{\text{'}\text{'}} \end{gathered}$$
$$\begin{gathered} =57°16\text{'}21.81\text{'}\text{'} \\ \approx 57°16\text{'}22\text{'}\text{'} \end{gathered}$$