Trigonometry: Converting Degrees to Radians (and vice versa)

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1 of 2 videos by Darshan Senthil

## Key Questions

• Imagine a circle and a central angle in it. If the length of an arc that this angle cuts off the circle equals to its radius, then, by definition, this angle's measure is 1 radian. If an angle is twice as big, the arc it cuts off the circle will be twice as long and the measure of this angle will be 2 radians. So, the ratio between an arc and a radius is a measure of a central angle in radians.

For this definition of the angle's measure in radians to be logically correct, it must be independent of a circle.
Indeed, if we increase the radius while leaving the central angle the same, the bigger arc that our angle cuts from a bigger circle will still be in the same proportion to a bigger radius because of similarity, and our measure of an angle will be the same and independent of a circle.

Since the length of a circumference of a circle equals to its radius multiplied by $2 \pi$, the full angle of ${360}^{0}$ equals to $2 \pi$ radians.

From this we can derive other equivalencies between degrees and radians:

${30}^{0} = \frac{\pi}{6}$
${45}^{0} = \frac{\pi}{4}$
${60}^{0} = \frac{\pi}{3}$
${90}^{0} = \frac{\pi}{2}$
${180}^{0} = \pi$
${270}^{0} = 3 \frac{\pi}{2}$
${360}^{0} = 2 \pi$

• Since

${180}^{\circ} = \pi$ radians,

if you want to convert $x$ degrees to radians, then

$x \times \frac{\pi}{180}$ radians,

and if you want to convert $x$ radians to degrees, then

$x \times \frac{180}{\pi}$ degrees

I hope that this was helpful.

Throug the equivalence pi=180Âº and de direct proportionality of arcs and angles. See examples

#### Explanation:

If we have 45Âº degrees convert to radians is easy

$\frac{\pi}{x} = \frac{180}{45}$. Then $x = \frac{45 \pi}{180} = \frac{\pi}{4}$. So 45Âº=pi/4 rad

By the other hand, if we have $\frac{\pi}{3}$ rads, to convert to degrees, we use the same equivalence

$\frac{\pi}{\frac{\pi}{3}} = \frac{180}{y}$. Trasposing terms y=(180pi)/(3pi)=180/3=60Âº

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