Question

How do you find the complement and supplement of $ \dfrac{\pi }{{12}} $ ?

Answer 1

Hint: Always remember that the sum of two complementary angles is always ninety degrees and the sum of two supplementary angles is one hundred and eighty degrees. Here we will use the properties of the complementary angle and then will place it in the given expression and will simplify for the required solution.

Complete step-by-step answer:
Let us assume that the given angle is “A” and the required complementary angle be “B”
 $ A = \dfrac{\pi }{{12}} $
For Complementary angles:
 $ A + B = \dfrac{\pi }{2} $
It can be re-written as –
 $ \Rightarrow B = \dfrac{\pi }{2} - \dfrac{\pi }{{12}} $
Take LCM (least common multiple)
 $
   \Rightarrow B = \dfrac{{6\pi }}{{12}} - \dfrac{\pi }{{12}} \\
   \Rightarrow B = \dfrac{{6\pi - \pi }}{{12}} \;
  $
Simplify the above expression –
 $ \Rightarrow B = \dfrac{{5\pi }}{{12}} $ ….. (I)
Now, let us assume “C” be the required supplementary angle –
 $ A + C = \pi $
It can be re-written as –
 $ \Rightarrow C = \pi - \dfrac{\pi }{{12}} $
Take LCM (least common multiple)
 $
   \Rightarrow C = \dfrac{{12\pi }}{{12}} - \dfrac{\pi }{{12}} \\
   \Rightarrow C = \dfrac{{12\pi - \pi }}{{12}} \;
  $
Simplify the above expression –
 $ \Rightarrow C = \dfrac{{11\pi }}{{12}} $ ….. (II)
Equations (I) and (II) are the required solutions.

Note: Always remember the difference between the complementary and supplementary angles. The sum of two angles in supplementary is always one-eighty degree. The most important property of sines and cosines is that their values lie between minus one and plus one. Every point on the circle is unit circle from the origin. So, the coordinates of any point are within one of zero as well.
Directly the Pythagoras identity are followed by sines and cosines which concludes that –
 $ si{n^2}\theta + co{s^2}\theta = 1 $