# Angles

## Angles

The main things to remember about angles is that they are measured starting from the positive x-axis, unless otherwise stated, and measured in a counter-clockwise direction.

We say that a whole revolution is 360 degrees, making an arrow pointing north a 90 degree angle, west a 180 angle, and south a 270 degree angle.

If we have an angle in degrees and we want the angle measure in radians we can use the conversion facto of ${\displaystyle 180^{\circ }=\pi {\text{ radians}}}$ This means that if we want to convert back and forth ${\displaystyle 1{\text{ degree}}={\frac {\pi }{180}}{\text{ radians}}\qquad 1{\text{ radian }}={\frac {180}{\pi }}{\text{ degrees}}}$

## Examples

Convert the following from degrees to radians or radians to degrees. It is usually understood that if an angle has a ${\displaystyle \pi }$ in it the angle measure is given in radians.

${\displaystyle a)\,245^{\circ }}$

${\displaystyle b)\,{\dfrac {5\pi }{4}}}$

${\displaystyle c)\,275^{\circ }}$

${\displaystyle d)\,{\dfrac {9\pi }{2}}}$

${\displaystyle e)\,30^{\circ }}$

Solutions To convert from degrees to radians, we multiply the degrees by ${\displaystyle {\dfrac {\pi }{180}}}$. To convert in the other direction, radians to degrees, we multiply the radians by ${\displaystyle {\dfrac {180}{\pi }}}$ ${\displaystyle a){\dfrac {245\pi }{180}}={\dfrac {49\cdot 5\pi }{36\cdot 5}}={\dfrac {49\pi }{36}}}$

${\displaystyle b){\dfrac {5\pi }{4}}\cdot {\dfrac {180}{\pi }}={\dfrac {180\cdot 5}{4}}={\dfrac {4\cdot 45\cdot 5}{4}}=45\cdot 5=225}$

${\displaystyle c){\dfrac {275\pi }{180}}={\dfrac {55\cdot 5\pi }{36\cdot 5}}={\dfrac {55\pi }{36}}}$

${\displaystyle d){\dfrac {9\pi }{2}}\cdot {\dfrac {180}{\pi }}=90\cdot 9=810}$

${\displaystyle e){\dfrac {30\pi }{180}}={\dfrac {\pi }{6}}}$

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