Ejercicio resuelto de Razones trigonométricas · Matemáticas I

Completar en el cuaderno la siguiente tabla:

$\overline{)\begin{array}{cccccccc}\text{Grados}& {105}^{\circ }& & {225}^{\circ }& & {320}^{\circ }& & {35}^{\circ }\\ \text{Radianes}& & {\displaystyle \frac{4\pi }{9}}\text{ rad}& & {\displaystyle \frac{\pi }{15}}\text{ rad}& & 1\text{ rad}& \end{array}}$

Se usa la relacion entre grados sexagesimales y radianes a partir de la equivalencia ${180}^{\circ }=\pi \text{ rad}$, de donde se obtienen las dos formulas de conversion:

$\text{radianes}=\text{grados}\cdot {\displaystyle \frac{\pi }{{180}^{\circ }}}\text{grados}=\text{radianes}\cdot {\displaystyle \frac{{180}^{\circ }}{\pi }}$

Convertimos a radianes los angulos dados en grados.

Para ${105}^{\circ }$:
${105}^{\circ }\cdot {\displaystyle \frac{\pi }{{180}^{\circ }}}={\displaystyle \frac{105\pi }{180}}={\displaystyle \frac{7\pi }{12}}\text{ rad}$

Para ${225}^{\circ }$:
${225}^{\circ }\cdot {\displaystyle \frac{\pi }{{180}^{\circ }}}={\displaystyle \frac{225\pi }{180}}={\displaystyle \frac{5\pi }{4}}\text{ rad}$

Para ${320}^{\circ }$:
${320}^{\circ }\cdot {\displaystyle \frac{\pi }{{180}^{\circ }}}={\displaystyle \frac{320\pi }{180}}={\displaystyle \frac{16\pi }{9}}\text{ rad}$

Para ${35}^{\circ }$:
${35}^{\circ }\cdot {\displaystyle \frac{\pi }{{180}^{\circ }}}={\displaystyle \frac{35\pi }{180}}={\displaystyle \frac{7\pi }{36}}\text{ rad}$

Convertimos a grados los angulos dados en radianes.

Para ${\displaystyle \frac{4\pi }{9}}\text{ rad}$:
${\displaystyle \frac{4\pi }{9}}\cdot {\displaystyle \frac{{180}^{\circ }}{\pi }}={\displaystyle \frac{4\cdot {180}^{\circ }}{9}}={\displaystyle \frac{{720}^{\circ }}{9}}={80}^{\circ }$

Para ${\displaystyle \frac{\pi }{15}}\text{ rad}$:
${\displaystyle \frac{\pi }{15}}\cdot {\displaystyle \frac{{180}^{\circ }}{\pi }}={\displaystyle \frac{{180}^{\circ }}{15}}={12}^{\circ }$

Para $1\text{ rad}$:
$1\cdot {\displaystyle \frac{{180}^{\circ }}{\pi }}={\displaystyle \frac{{180}^{\circ }}{\pi }}\approx {\mathrm{57,2958}}^{\circ }$

Expresamos este resultado en grados, minutos y segundos. La parte decimal de grados se pasa a minutos multiplicando por $60$, y la parte decimal de minutos se pasa a segundos multiplicando de nuevo por $60$:
${\mathrm{0,2958}}^{\circ }\cdot 60={\mathrm{17,75}}^{\mathrm{\prime }}$
${\mathrm{0,75}}^{\mathrm{\prime }}\cdot 60={45}^{\mathrm{\prime }\mathrm{\prime }}$

Por tanto:
$1\text{ rad}\approx {57}^{\circ }{17}^{\mathrm{\prime }}{45}^{\mathrm{\prime }\mathrm{\prime }}$

La tabla completa queda:

$\overline{)\begin{array}{cccccccc}\text{Grados}& {105}^{\circ }& {80}^{\circ }& {225}^{\circ }& {12}^{\circ }& {320}^{\circ }& {57}^{\circ }{17}^{\mathrm{\prime }}{45}^{\mathrm{\prime }\mathrm{\prime }}& {35}^{\circ }\\ \text{Radianes}& {\displaystyle \frac{7\pi }{12}}& {\displaystyle \frac{4\pi }{9}}& {\displaystyle \frac{5\pi }{4}}& {\displaystyle \frac{\pi }{15}}& {\displaystyle \frac{16\pi }{9}}& 1& {\displaystyle \frac{7\pi }{36}}\end{array}}$