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.

Radians

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

Examples

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

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

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

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

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

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

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

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

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

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

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

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