Question

How do you use the sum to product formula to write the sum or difference $\sin 3\theta +\sin \theta $ as a product?

Answer 1

Hint: In this question we have been given a sum of trigonometric expressions which we have to evaluate and convert into its respective product without changing its value. We have been given the expression of sum of two sine functions therefore, we will use the sum to product formula $\sin X+\sin Y=2\sin \left( \dfrac{X+Y}{2} \right)\cos \left( \dfrac{X-Y}{2} \right)$ and substitute the angles and simplify to get the required expression in product format.

Complete step by step solution:
We have the expression as:
$\Rightarrow \sin 3\theta +\sin \theta $
We can see that the expression is in the form of $\sin X+\sin Y$, where $X=3\theta $ and $Y=\theta $ therefore, we will use the sum to product formula $\sin X+\sin Y=2\sin \left( \dfrac{X+Y}{2} \right)\cos \left( \dfrac{X-Y}{2} \right)$.
On substituting the values in the formula, we get:
$\Rightarrow 2\sin \left( \dfrac{3\theta +\theta }{2} \right)\cos \left( \dfrac{3\theta -\theta }{2} \right)$
On adding and subtracting the angles, we get:
$\Rightarrow 2\sin \left( \dfrac{4\theta }{2} \right)\cos \left( \dfrac{2\theta }{2} \right)$
On dividing the angles in the function, we get:
$\Rightarrow 2\sin 2\theta \cos \theta $
Now we have in the expression the sine function of double angle. We know that $\sin \left( 2\theta \right)=2\sin \theta \cos \theta $ therefore, on substituting it in the formula, we get:
$\Rightarrow 2\left( 2\sin \theta \cos \theta \right)\cos \theta $
On simplifying the brackets and multiplying, we get:
$\Rightarrow 4\sin \theta {{\cos }^{2}}\theta $, which is the required solution.

Note: It is to be remembered that when a trigonometric expression is converted from its sum form to the product form, the value of the function does not change. Trigonometric terms in subtraction can also be converted into their products using the subtraction formula. For example, the trigonometric expression $\sin X-\sin Y$ can be converted into the product form using the formula $\sin X-\sin Y=2\cos \left( \dfrac{x+y}{2} \right)\sin \left( \dfrac{x-y}{2} \right)$. There also exist formulas for converting various other trigonometric identities like cosine, tangent etc.