Problem 61 - Precalculus Mathematics for Calculus

Question

Write the sum as a product. $$\sin 5 x+\sin 3 x$$

Step-by-Step Solution

Answer

Use the identity: \( \sin 5x + \sin 3x = 2 \sin 4x \cos x \).

Step 1: Recall the Trigonometric Identity

Recall the sum-to-product identity for sine: \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \). This identity is useful for transforming a sum of two sines into a product.

Step 2: Identify A and B

In our expression \(\sin 5x + \sin 3x\), we identify \(A = 5x\) and \(B = 3x\).

Step 3: Apply the Sum-to-Product Formula

Using the identity from Step 1, substitute \(A = 5x\) and \(B = 3x\) into it:\[\sin 5x + \sin 3x = 2 \sin\left(\frac{5x+3x}{2}\right) \cos\left(\frac{5x-3x}{2}\right).\]

Step 4: Simplify the Expression

Calculate \(\frac{5x+3x}{2} = 4x\) and \(\frac{5x-3x}{2} = x\). Substitute these values back into the formula:\[\sin 5x + \sin 3x = 2 \sin 4x \cos x.\]

Step 5: Verify the Result

Ensure that the transformation maintains equality by checking the identity or simplifying back into the original form, if needed. This confirms that the expression has been correctly rewritten as a product.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are a set of trigonometric formulas that allow us to transform the sum or difference of two trigonometric functions into a product of trigonometric functions. This transformation is extremely helpful, especially in simplifying complex trigonometric expressions and solving equations. One of the most common sum-to-product identities deals with sine and is expressed as:

• \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)

This identity essentially takes two sine functions and rewrites them as a product involving sine and cosine. The variables \( A \) and \( B \) represent the angles of the sine functions.
By applying this identity, you can simplify the sum of sines which is often beneficial for integration or further algebraic manipulation. Whenever you have an expression that resembles \( \sin A + \sin B \), remember this identity for a potentially simpler form.

Sine Function

The sine function is one of the basic trigonometric functions and plays an important role in mathematics, physics, and engineering. It is a periodic function with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units. The sine function, written as \( \sin(x) \), represents the ratio of the opposite side to the hypotenuse in a right triangle, making it significant in the study of triangles.

• The graph of the sine function is a smooth, wave-like curve, often referred to as a sinusoidal wave.
• Its values range from -1 to 1.
• Sine is an odd function, so \( \sin(-x) = -\sin(x) \).

A deep understanding of the sine function and its properties, like periodicity and symmetry, is essential for efficiently working with trigonometric identities and transformations such as the sum-to-product identities.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra and trigonometry that involves rewriting expressions in a simpler or more compact form. This process makes it easier to perform further calculations and solve mathematical problems. In trigonometry, simplifying an expression often requires the application of various identities, such as the sum-to-product identities.In the context of the exercise given, simplifying involves transforming the expression \( \sin 5x + \sin 3x \) using a known identity:

• The expression \( \sin 5x + \sin 3x \) can be rewritten as \( 2 \sin 4x \cos x \).

This simplification not only makes the expression more concise but also prepares it for further operations, such as integrations, which are more straightforward in product form.
It's important to practice these kinds of transformations to build confidence and efficiency in working with more complex trigonometric expressions and equations.