Question

Evaluate $\sin {12^o}\sin {24^o}\sin {48^o}\sin {84^o} = $
A. $1$
B. $\dfrac{1}{{16}}$
C. $\dfrac{3}{{15}}$
D. None

Answer 1

Hint: Multiply and divide by 4 and use a sine formula accordingly to finish solving.
First, we multiply and divide by 4 and then bring them in such a form that we can use a $2\sin a\sin b$ so, that it can be calculated further, we should make the $\sin a\sin b$ combination such that the further formed angles after applying the formula are known pre hand. Like $\sin {12^0}\sin {48^o}$ such that 60 degrees is obtained. Then on substituting the angle values and the simplified calculation, we get the answer for the problem.

Complete step by step solution:
First multiply and divide by 4.
$\dfrac{4}{4}\sin {12^o}\sin {24^o}\sin {48^o}\sin {84^o}$
Then we divide the 4 in the numerator as a product of 2, such that it can be used to form $2\sin a\sin b$ and also form a pair such that the further calculation is easy.
$\dfrac{1}{4}\left( {2\sin {{12}^o}\sin {{48}^o}} \right)\left( {2\sin {{24}^o}\sin {{84}^o}} \right)$
Now, that we have obtained the form, we apply the formula
$2\sin a\sin b = \cos (a - b) - \cos (a + b)$
So, we get
\[
  \dfrac{1}{4}\left( {\cos (48 - 12) - \cos (48 + 12)} \right)\left( {\cos (84 - 24) - \cos (84 + 24)} \right) \\
  \dfrac{1}{4}(\cos 36 - \cos 60)(\cos 60 - \cos 108) \\
\]
From the above we convert$\cos 108$ in terms of sine by writing it in terms of \[\cos (90 + 18)\].
And in the second quadrant, cosine is negative, so we replace $\cos 108$ with $ - \sin 18$. Then we get as,
$\dfrac{1}{4}(\cos 36 - \cos 60)(\cos 60 + \sin 18)$
Now, since we know the values of all the angles, we can replace them and get the answer.
$\dfrac{1}{4}\left( {\dfrac{{\sqrt 5 + 1}}{4} - \dfrac{1}{2}} \right)\left( {\dfrac{1}{2} + \dfrac{{\sqrt 5 - 1}}{4}} \right)$
Now, we split the numerator with the denominator 4
$
   = \dfrac{1}{4}\left( {\dfrac{{\sqrt 5 }}{4} + \dfrac{1}{4} - \dfrac{1}{2}} \right)\left( {\dfrac{1}{2} + \dfrac{{\sqrt 5 }}{4} - \dfrac{1}{4}} \right) \\
   = \dfrac{1}{4}\left( {\dfrac{{\sqrt 5 }}{4} - \dfrac{1}{4}} \right)\left( {\dfrac{{\sqrt 5 }}{4} + \dfrac{1}{4}} \right) \\
   = \dfrac{1}{4}\left( {\dfrac{{(\sqrt 5 - 1)(\sqrt 5 + 1)}}{4}} \right) \\
$
It is of the form $(a - b)(a + b)$ which gives $({a^2} - {b^2})$
$
   = \dfrac{1}{{64}}(5 - 1) \\
   = \dfrac{4}{{64}} \\
   = \dfrac{1}{{16}} \\
$
So, the correct answer is Option B.

Note: For problems like this, it is necessary to know the absolute value of some angles like 18 degree and 36 degree. As it makes it easy and saves time during the calculation and solving. Also, this cannot be the only way to solve but it is the easiest way to solve the problem.