Problem 25 - Precalculus: Functions and Graphs

Question

Express as a sum. $$(\sin a x)(\cos b x)$$

Step-by-Step Solution

Answer

\(\frac{1}{2} (\sin((a+b)x) + \sin((a-b)x))\).

Step 1: Identify the Trigonometric Identity

We begin by identifying the appropriate trigonometric identity that allows us to write the product of two trigonometric functions as a sum. The identity is:\[ \sin A \cos B = \frac{1}{2} \left(\sin(A + B) + \sin(A - B) \right) \] where \(A = ax\) and \(B = bx\).

Step 2: Substitute into Identity

Substitute \(A = ax\) and \(B = bx\) into the identity:\[ \sin(ax) \cos(bx) = \frac{1}{2} \left( \sin((a + b)x) + \sin((a - b)x) \right) \]

Step 3: Final Expression

After substituting, the final expression for \((\sin ax)(\cos bx)\) as a sum is:\[ \sin(ax) \cos(bx) = \frac{1}{2} \left( \sin((a+b)x) + \sin((a-b)x) \right) \]

Key Concepts

Product-to-Sum Formulas

The product-to-sum formulas are special trigonometric identities that convert products of sine and cosine into sums. They help simplify and solve complex trigonometric expressions or integrals. One widely used product-to-sum formula is:\[ \sin A \cos B = \frac{1}{2} \left(\sin(A + B) + \sin(A - B) \right) \]These are particularly powerful when dealing with expressions involving trigonometric function products or simplifying integrals that involve such products. The main advantage of these identities is that they reduce the complexity of the problem, allowing us to work with sums of functions instead of their products.
To apply them, identify the coefficients and substitute them into the formula accordingly. This transformation makes these functions easier to manipulate for further solving.

Sine Function

The sine function, denoted as \(\sin(x)\), is one of the fundamental trigonometric functions. It is a periodic function with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

• The sine function ranges from -1 to 1.
• It is an odd function, satisfying \(\sin(-x) = -\sin(x)\).
• Its graph is a smooth, wave-like curve that crosses the origin.

In trigonometry, the sine function is often used to describe the y-coordinate of a point on the unit circle. It plays a critical role in identifying angles and solving triangles.
Moreover, in calculus and real-world applications, the sine function is used extensively in physics, engineering, and signal processing to model periodic phenomena such as sound and light waves.

Cosine Function

The cosine function, denoted as \(\cos(x)\), is another fundamental trigonometric function, closely related to sine. Like the sine function, cosine is periodic but differs in phase.

• The cosine function also ranges from -1 to 1.
• It is an even function, meaning \(\cos(-x) = \cos(x)\).
• Its period is \(2\pi\), repeating every \(2\pi\) radians.

The graph of the cosine function is similar to that of the sine function but starts at 1 when \(x = 0\) and has a similar wave-like pattern. In terms of the unit circle, the cosine of an angle is the x-coordinate of a point on the circle.
Cosine functions are pivotal in trigonometry for solving problems involving right triangles, and they also appear in various fields like signal processing and vibration analysis, providing a real-world context to their mathematical properties.