Question

What will be the value of $\sin {78^0} - \sin {18^0} + \sin {30^0} - \sin {42^0}$
$
  (a){\text{ }}\dfrac{3}{2} \\
  (b){\text{ }}\dfrac{1}{2} \\
  (c){\text{ }}\dfrac{5}{2} \\
  (d){\text{ None of these}} \\
 $

Answer 1

Hint – In this question rearrange the given expression such that the highest and the second highest angle are kept together, then apply the trigonometric identity $\sin C - \sin D = 2\sin \left( {\dfrac{{C - D}}{2}} \right)\cos \left( {\dfrac{{C + D}}{2}} \right)$ to simplify it further. This will help to get the right answer.

Complete step-by-step solution -
Given trigonometric expression is
$\sin {78^0} - \sin {18^0} + \sin {30^0} - \sin {42^0}$
Now rearrange the terms we have,
$ \Rightarrow \sin {78^0} - \sin {42^0} - \sin {18^0} + \sin {30^0}$
Now as we know that $\sin C - \sin D = 2\sin \left( {\dfrac{{C - D}}{2}} \right)\cos \left( {\dfrac{{C + D}}{2}} \right)$ so use this property in above equation we have,
$ \Rightarrow \sin {78^0} - \sin {42^0} - \sin {18^0} + \sin {30^0} = 2\sin \left( {\dfrac{{{{78}^0} - {{42}^0}}}{2}} \right)\cos \left( {\dfrac{{78 + 42}}{2}} \right) - \sin {18^0} + \sin {30^0}$
Now simplify the above equation we have,
$ \Rightarrow 2\sin \left( {\dfrac{{{{78}^0} - {{42}^0}}}{2}} \right)\cos \left( {\dfrac{{78 + 42}}{2}} \right) - \sin {18^0} + \sin {30^0} = 2\sin {18^0}\cos {60^0} - \sin {18^0} + \sin {30^0}$
Now as we know that $ \cos {60^0} $ = $\dfrac{1}{2} \text{and} \sin {30^0} = \dfrac{1}{2} $ so substitute these values in above equation we have,
$ \Rightarrow 2\sin {18^0}\cos {60^0} - \sin {18^0} + \sin {30^0} = 2 \times \dfrac{1}{2}\sin {18^0} - \sin {18^0} + \dfrac{1}{2}$
Now simplify the above equation we have,
$ \Rightarrow 2 \times \dfrac{1}{2}\sin {18^0} - \sin {18^0} + \dfrac{1}{2} = \sin {18^0} - \sin {18^0} + \dfrac{1}{2} = \dfrac{1}{2}$
So this is the required answer.
Hence option (B) is correct.

Note – It is always advisable to remember the value of some of the basic trigonometric ratios like $\cos {60^0} = \dfrac{1}{2} \text{and} \sin {30^0} = \dfrac{1}{2} $ along with some basic trigonometric identities. These questions are in general formula based only so having a good gist of identities helps saving a lot of time.