Question

The value of $\sin \left( {\dfrac{\pi }{{10}}} \right)\sin \left( {\dfrac{{3\pi }}{{10}}} \right)$ is equal to:
A. $\dfrac{1}{2}$
B. $- \dfrac{1}{2}$
C. $\dfrac{1}{4}$
D. $1$

Answer 1

The given question requires us to find the value of the product of two sine functions with different angles. So, we will do the basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $\sin \left( {{{90}^ \circ } - x} \right) = \cos x$ and double angle formula for sine. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. We must know the simplification rules to solve the problem with ease.

Complete step by step answer:
In the given problem, we have to find value of $\sin \left( {\dfrac{\pi }{{10}}} \right)\sin \left( {\dfrac{{3\pi }}{{10}}} \right)$.
So, we have, $\sin \left( {\dfrac{\pi }{{10}}} \right)\sin \left( {\dfrac{{3\pi }}{{10}}} \right)$
Using the trigonometric identity $\sin \left( {{{90}^ \circ } - x} \right) = \cos x$, we get,
$\Rightarrow \sin \left( {\dfrac{\pi }{{10}}} \right)\cos \left( {\dfrac{\pi }{2} - \dfrac{{3\pi }}{{10}}} \right)$
Simplifying the angle in cosine, we get,
$\Rightarrow \sin \left( {\dfrac{\pi }{{10}}} \right)\cos \left( {\dfrac{{5\pi - 3\pi }}{{10}}} \right)$
$\Rightarrow \sin \left( {\dfrac{\pi }{{10}}} \right)\cos \left( {\dfrac{{2\pi }}{{10}}} \right)$

Multiplying and dividing the expression by $2\cos \left( {\dfrac{\pi }{{10}}} \right)$, we get,
$\Rightarrow \dfrac{{2\cos \left( {\dfrac{\pi }{{10}}} \right)}}{{2\cos \left( {\dfrac{\pi }{{10}}} \right)}} \times \sin \left( {\dfrac{\pi }{{10}}} \right)\cos \left( {\dfrac{{2\pi }}{{10}}} \right)$
Now, using the double angle formula for sine $\sin 2x = 2\sin x\cos x$, we get,
$\Rightarrow \dfrac{{\sin \left( {\dfrac{{2\pi }}{{10}}} \right)}}{{2\cos \left( {\dfrac{\pi }{{10}}} \right)}} \times \cos \left( {\dfrac{{2\pi }}{{10}}} \right)$
Now, multiplying and dividing the expression by $2$, we get,
$\Rightarrow \dfrac{{2\sin \left( {\dfrac{{2\pi }}{{10}}} \right)\cos \left( {\dfrac{{2\pi }}{{10}}} \right)}}{{2 \times 2\cos \left( {\dfrac{\pi }{{10}}} \right)}}$
Now, using the double angle formula for sine again, we get,
$\Rightarrow \dfrac{{\sin \left( {\dfrac{{4\pi }}{{10}}} \right)}}{{4\cos \left( {\dfrac{\pi }{{10}}} \right)}}$

Now, we know that sine and cosine are complementary trigonometric functions. So, we get,
$\Rightarrow \dfrac{{\cos \left( {\dfrac{\pi }{2} - \dfrac{{4\pi }}{{10}}} \right)}}{{4\cos \left( {\dfrac{\pi }{{10}}} \right)}}$
Simplifying the expression further, we get,
$\Rightarrow \dfrac{{\cos \left( {\dfrac{{5\pi - 4\pi }}{{10}}} \right)}}{{4\cos \left( {\dfrac{\pi }{{10}}} \right)}}$
$\Rightarrow \dfrac{{\cos \left( {\dfrac{\pi }{{10}}} \right)}}{{4\cos \left( {\dfrac{\pi }{{10}}} \right)}}$
Cancelling the common factors in numerator and denominator, we get,
$\Rightarrow \dfrac{1}{4}$
So, the value of the trigonometric expression $\sin \left( {\dfrac{\pi }{{10}}} \right)\sin \left( {\dfrac{{3\pi }}{{10}}} \right)$ is $\dfrac{1}{4}$.

Hence, option C is the correct answer.

Note: We must have a strong grip over the concepts of trigonometry, related formulae and rules to ace these types of questions. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such type of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations.