Product-to-Sum Trigonometric Formulas
Product-to-sum trigonometric formulas can be very helpful in simplifying a trigonometric expression by taking the product term such as or and converting it into a sum.
Contents
Formulas
The complete list of product-to-sum trigonometric formulas are as follows:
Proof
The product-to-sum formulas can be obtained by observing that the sum and difference formulas for sine and cosine look very similar except for opposite signs in the middle. Then by combining the expressions, we can cancel terms.
We have
The right-hand side is a product of cosine terms, while the left-hand side is a sum of cosine terms.
Using and identities, we have
Therefore,
Applications
Evaluate .
We know that Substituting and , we get
Express as a sum of basic trigonometric functions (the solution should not include any products of trigonometric functions).
The product-to-sum formulas can be helpful in solving integration problems involving the product of trigonometric ratios.
Integrate
This problem may seem tough at first, but after using the product-to-sum trigonometric formula, this integral very quickly changes into a standard form.
Using the first formula stated above, the integral is equivalent to
\[\begin{align}
\phantom{=} \int \frac{1}{2} \big( \sin (3x - 4x) + \sin(3x + 4x) \big) \, \mathrm{d}x & = \int \frac{1}{2} \big( \sin (-x) + \sin(7x) \big) \, \mathrm{d}x\\ & = \frac{1}{2} \left( \cos x - \frac{\cos 7x}{7}\right)+C, \end{align}\]where is the constant of integration.