Problems from Angles in radians

Development:

1. We know that the sum of all angles in a triangle is $180t{h}^{\circ }$, since an equilateral triangle has three equal angles what we must write is ${60}^{\circ }$. Let's now convert this into radians by means of the conversion factor that transforms from degrees to radians: $${60}^{\circ }\cdot {\displaystyle \frac{2\pi \text{radians}}{{360}^{\circ }}}={\displaystyle \frac{60\cdot 2\pi }{360}}\text{radians}={\displaystyle \frac{\pi }{3}}\text{radians}$$

2. We know that a full rotation is ${360}^{\circ }$, therefore three full rotations will be $3\cdot {360}^{\circ }$ giving a total of ${1080}^{\circ }$. But, on the other hand, we also know that a full rotation corresponds to the total longitude of the circumference which, in this case, is $2\pi$. If there are three turns, there are $3\cdot 2\pi$ which equals $6\pi$ radians. If we prefer to do it by means of a conversion factors then: $${1080}^{\circ }\cdot {\displaystyle \frac{2\pi \text{radians}}{{360}^{\circ }}}={\displaystyle \frac{1080\cdot 2\pi }{360}}\text{radians}=6\pi \text{radians}$$

Solution:

1. ${60}^{\circ }={\displaystyle \frac{\pi }{3}}\text{radians}$
2. ${1080}^{\circ }=6\pi \text{radians}$

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