Question

Express the following sexagesimal measure as radian measure and centesimal system
${135}^o$
${270}^o$
${72}^o$

Answer 1

$\left(i\right)$We know that ${180}^{\circ }={\pi }^c$

$\Rightarrow {135}^{\circ }=\frac{{135}^{\circ }\times {\pi }^c}{{180}^{\circ }}$

$=\frac{135}{180}\times \frac{22}{7}$ where $\pi =\frac{22}{7}$

$=\frac{27}{36}\times \frac{22}{7}={\frac{33}{14}}^c=2.356194$radians

We know that $400$gradian $={360}^{\circ }$

$\Rightarrow {135}^{\circ }=\frac{135}{400}\times {360}^{\circ }=121.5$ gradians

$\left(ii\right)$We know that ${180}^{\circ }={\pi }^c$

$\Rightarrow {270}^{\circ }=\frac{{270}^{\circ }\times {\pi }^c}{{180}^{\circ }}$

$=\frac{270}{180}\times \pi$

$=\frac{27\pi }{18}={\frac{3\pi }{4}}^c$

We know that $1$gradian $={400}^g={360}^{\circ }$

$\Rightarrow {270}^{\circ }=\frac{270}{360}\times 400=300$ gradians