Question

Find the value of the given equation \[\sin \left( {\dfrac{\pi }{{12}}} \right)\] using sum and difference identity?

Answer 1

Hint: Sum and difference identity in trigonometry implies the expansion of the given identity with the given sum or difference of the angle, like in algebra we have expansion for the general identity in the same way in trigonometry also we have to follow the rules and solve accordingly.
Formulae Used: \[\sin \left( {a - b} \right) = \sin a\sin b - \cos a\cos b\] , \[\sin \dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2},\,\sin \dfrac{\pi }{4} = \cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }},\,\cos \dfrac{\pi }{3} = \dfrac{1}{2}\]

Complete step-by-step answer:
For the given trigonometric equation \[\sin \left( {\dfrac{\pi }{{12}}} \right)\]
We can break the given angles as
 \[ \Rightarrow \dfrac{\pi }{{12}} = \left( {\dfrac{\pi }{3} - \dfrac{\pi }{4}} \right)\]
To check that above breaking of angle is correct or not we can solve the R.H.S, on solving we get;
 \[ \Rightarrow \left( {\dfrac{\pi }{3} - \dfrac{\pi }{4}} \right) = \dfrac{{4\pi - 3\pi }}{{12}} = \dfrac{\pi }{{12}}\left( {L.C.M\,of\,3\,and\,4\,is\,12} \right)\]
Hence our breaking of angle is correct.
Now on further solving and using the trigonometric identity that is:
 \[ \Rightarrow \sin \left( {a - b} \right) = \sin a\sin b - \cos a\cos b\]
On comparing our equation with the general equation we get;
 \[ \Rightarrow a = \dfrac{\pi }{3},\,b = \dfrac{\pi }{4}\]
Now on solving our equation we get:
 \[
   \Rightarrow \sin \left( {\dfrac{\pi }{3} - \dfrac{\pi }{4}} \right) = \sin \dfrac{\pi }{3}\sin \dfrac{\pi }{4} - \cos \dfrac{\pi }{3}\cos \dfrac{\pi }{4} \\
= \left( {\dfrac{{\sqrt 3 }}{2} \times \dfrac{1}{{\sqrt 2 }}} \right) - \left( {\dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }}} \right) \\
= \dfrac{{\sqrt 3 }}{{2\sqrt 2 }} - \dfrac{1}{{2\sqrt 2 }} = \dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }} \;
\]
 This is our required answer.
So, the correct answer is “$\dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}$”.

Note: While dealing with bigger angles you should always be aware for the breaking of angles in smaller forms because only smaller angles starting from zero, thirty, forty five, sixty and ninety can be easily remembered and solution can be easily obtained.
Every trigonometric identity has a rule while conversion to its smaller form and you should all convert the angles easily, rules state the plus and minus sign after conversion the angle. While breaking the angle you should know how to break according to our convenience, always try to break in such a way that broken angle value can easily be determined or already known.