Problem 15 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \sin x+\sin 2 x $$

Step-by-Step Solution

Answer

The sum of the given sine functions can be expressed as a product: \( \sin x (1 + 2 \cos x) \)

Step 1: Identify relevant formulas

The relevant trigonometric identity to solve this exercise is \( \sin(p+q) = \sin p \cos q + \cos p \sin q \). In this case, \( p = x \) and \( q = x \) to form \( 2x \) for the second term.

Step 2: Arrange terms

Break the second term \( \sin 2x \) into two parts using the identity from Step 1. So, \( \sin x + \sin 2x \) becomes \( \sin x + 2 \sin x \cos x \).

Step 3: Simplify the result

Factor out the term \( \sin x \) from each term to get \( \sin x (1 + 2 \cos x) \).

Key Concepts

Sum to Product Formulas

Sum to product formulas are special equations used in trigonometry to transform sums or differences of trigonometric terms into products. These identities are particularly useful in simplifying expressions and solving trigonometric equations.

In this exercise, we are working with this principle to express the sum of two sine functions as a product.

• For example, the sum formula for sine is: \[\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)\]
• This approach helps simplify problems by breaking down complex expressions into more manageable parts.

In our original problem, \[\sin x + \sin 2x\]we use these identities to rewrite the expression in a way that factors can easily be identified, helping us to find a product representation. This method not only simplifies calculation steps but also aids in understanding deeper trigonometric relationships.

Sine Function

The sine function is a fundamental concept in trigonometry, representing the ratio of the opposite side to the hypotenuse in a right triangle. It's essential in various fields including physics, engineering, and mathematics.

The function itself oscillates between -1 and 1, creating a wave-like pattern that's periodic every \(2\pi\). This oscillation makes sine a key player in wave-related phenomena such as sound and light. When dealing with sums of sines, as in the problem \(\sin x + \sin 2x\), we dive into identities that allow transformation into more usable forms.

• \(\sin 2x\) can be expanded using the double angle identity: \[\sin 2x = 2 \sin x \cos x\]
• Incorporating this into our expression is crucial for simplification.

This reveals the sum's nature, showing how trigonometric identities can convert and simplify problems while maintaining their core properties.

Factoring Expressions

Factoring expressions is a mathematical technique used to simplify complex expressions by breaking them down into products of simpler factors. In trigonometry, factoring involves using identities to extract common factors from trigonometric expressions.

This step is key when you already have an expanded form and aim to condense the expression into more digestible terms.In our exercise, the expression \[\sin x + 2 \sin x \cos x\]goes through a factoring process. We identify that \(\sin x\) is a common factor that can be factored out:

• This transforms the expression to: \[\sin x (1 + 2 \cos x)\]
• This form is much simpler and highlights the product of two expressions, showcasing the effectiveness of factoring.

Factoring is not just about simplifying; it allows us to see hidden details of an equation or expression, offering insights into their underlying structures.