91Ó°ÊÓ

The product-to-sum formula is a mathematical tool used to simplify the integration of products of trigonometric functions, like sines and cosines. It helps us convert the product of two trigonometric functions into a sum, which can make integration much easier. In this exercise, we dealt with a product of cosines:

• \( \cos A \cos B = \frac{1}{2}(\cos(A+B) + \cos(A-B)) \)

To apply this formula, plug in the appropriate angle values. For example:

• Let \( A = \frac{\theta}{3} \) and \( B = \frac{\theta}{4} \)

When you substitute these values, the expression becomes \( \frac{1}{2} \left( \cos \frac{7\theta}{12} + \cos \frac{\theta}{12} \right) \). This conversion from product to sum assists in breaking down complex trigonometric integrals into simpler forms.

Trigonometric integrals involve the integration of expressions containing trigonometric functions, such as sine, cosine, tangent, etc. Techniques like the product-to-sum formula play a crucial role in simplifying these integrals.
In our example, by transforming \( \cos \frac{\theta}{3} \cos \frac{\theta}{4} \) using the product-to-sum formula, we ended up with two separate cosine terms. This leads to two simpler integrals:

• \( \int \cos \frac{7\theta}{12} \, d\theta \)
• \( \int \cos \frac{\theta}{12} \, d\theta \)

Each trigonometric integral can then be solved individually, employing standard integral rules for cosine, specifically the integral of \( \cos(k\theta) \). This is a common technique that makes solving intricate integrals more approachable.

Tables of integrals are references in calculus that list common integrals of functions, allowing for quick lookup of common integral forms. These provide pre-solved antiderivatives for various functions, including trigonometric functions like cosine and sine.
In our exercise, once we converted the original trigonometric product into a sum using the product-to-sum formula, we could look up the individual integral forms in a standard table of integrals. The integral of \( \cos(k\theta) \) is given by:

• \( \int \cos(k\theta) \, d\theta = \frac{1}{k} \sin(k\theta) + C \)

Utilizing this result, each part of the sum from the product-to-sum formula was integrated separately. For example:

• \( \int \cos \frac{7\theta}{12} \, d\theta = \frac{12}{7} \sin \frac{7\theta}{12} + C_1 \)
• \( \int \cos \frac{\theta}{12} \, d\theta = 12\sin \frac{\theta}{12} + C_2 \)

The table of integrals streamlines solving such trigonometric integrals by offering direct solutions, saving time and effort in computation.