Problems from Angles in radians

  1. Write in degrees and in radians the range of an angle of any equilateral triangle.
  2. Write in degrees and radians three full rotations to the circumference unit.
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  1. We know that the sum of all angles in a triangle is $$180th^\circ$$, 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: $$$60^\circ \cdot \dfrac{2\pi \ \mbox{radians}}{360^\circ}=\dfrac{60\cdot 2\pi}{360}\mbox{radians} = \dfrac{\pi}{3} \mbox{radians} $$$

  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: $$$1080^\circ \cdot \dfrac{2\pi \ \mbox{radians}}{360^\circ}=\dfrac{1080\cdot 2\pi}{360}\mbox{radians} = 6\pi \ \mbox{radians} $$$


  1. $$60^\circ = \dfrac{\pi}{3} \mbox{radians} $$
  2. $$1080^\circ = 6\pi \ \mbox{radians} $$
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