Problem 40 - Precalculus with Calculus Previews

Question

If \(x_{1}+x_{2}+x_{3}=\pi\), then show that \(\sin 2 x_{1}+\sin 2 x_{2}+\sin 2 x_{3}=4 \sin x_{1} \sin x_{2} \sin x_{3}\)

Step-by-Step Solution

Answer

Trigonometric identities and symmetry reduce the sin sums to a product form.

Step 1: Identify the Given Equation

We are given that \(x_1 + x_2 + x_3 = \pi\). This equation is fundamental for further transformations and applying trigonometric identities.

Step 2: Apply the Angle Sum Identity for Sines

The goal is to express \(\sin 2x_1 + \sin 2x_2 + \sin 2x_3\) using the angle sum identity. The identity \(\sin(A + B + C) = \sin A \cos(B+C) + \sin(B+C) \cos A\) can be utilized here.

Step 3: Simplify Using Trigonometric Identities

First convert \(\sin 2x_i\) into \(2 \sin x_i \cos x_i\). Therefore, \(\sin 2x_1 + \sin 2x_2 + \sin 2x_3 = 2(\sin x_1 \cos x_1 + \sin x_2 \cos x_2 + \sin x_3 \cos x_3)\).

Step 4: Transform Using the Given Condition

Use \(x_1 + x_2 + x_3 = \pi\) to imply that \(\sin(x_3) = \sin(\pi - (x_1 + x_2)) = \sin(x_1 + x_2)\). This is based on the identity \(\sin(\pi - \theta) = \sin \theta\).

Step 5: Use Product-to-Sum Formulas

Find \(\sin(x_1 + x_2)\) as \(\sin x_1 \cos x_2 + \cos x_1 \sin x_2\). Hence, \(\sin 2x_3 = \sin(2(x_1 + x_2)) = 2 \sin(x_1 + x_2) \cos(x_1 + x_2)\).

Step 6: Equate and Simplify

Combine and equate terms. Rewriting \(2(\sin x_1 \cos x_2 + \sin x_2 \cos x_1 + \sin x_3 \cos x_3)\) must equal \(4 \sin x_1 \sin x_2 \sin x_3\), validating through symmetry or identity expansions.

Key Concepts

Sum of Angles

When working with problems in trigonometry, the "sum of angles" identity is a valuable tool. This particular problem gave us the condition that the sum of three angles \( x_1, x_2, \) and \( x_3 \) is equal to \( \pi \). This condition is essential as it dictates how the angles relate to one another.
Understanding that \( x_1 + x_2 + x_3 = \pi \) means the angles form a semi-circle. Therefore, depending on the components \( x_1 \) and \( x_2 \), \( x_3 \) will adjust to always ensure their sum equals \( \pi \).

• Using this relation, we can employ identities such as \( \sin(\pi - \theta) = \sin(\theta) \), which means that the sine of an angle is the same as the sine of its supplementary angle.
• This relationship helps simplify complex trigonometric expressions by reducing them into more manageable terms.

Applying these identities is pivotal in transforming or simplifying trigonometric expressions. Recognizing the given condition \( x_1 + x_2 + x_3 = \pi \) is the first step in understanding how the trigonometric components of the equation will interact.

Product-to-Sum Formulas

To progress through this exercise, we utilize the product-to-sum formulas. These formulas are exceptionally helpful when simplifying or transforming trigonometric expressions.
The relevant product-to-sum identity used here helps in converting product terms into a sum or difference of trigonometric functions. For example, transforming expressions such as \( \sin u \sin v \) or \( \sin u \cos v \) into manageable forms:

• \( \sin u \cos v = \frac{1}{2} [\sin(u+v) + \sin(u-v)] \)
• By applying these formulas, complex expressions initially presented in terms of products can be simplified into sums, which are often easier to manipulate or interpret.

In this particular problem, the formula serves in expressing \( \sin(x_1 + x_2) \) and \( \sin(2(x_1 + x_2)) \) in such a way that aligns with our established condition, \( x_1 + x_2 + x_3 = \pi \). This transformation is a key step in ultimately showing the equivalence proposed by the problem statement.

Trigonometric Simplification

Trigonometric simplification is a process of transforming complex trigonometric expressions into simpler or more convenient forms.
In this exercise, simplifying trigonometric forms begins with acknowledging essential identities and relationships, like \( \sin 2x = 2 \sin x \cos x \). This is pivotal in breaking down expressions into components that can be further simplified or equated.

• For instance, \( \sin 2x_1 + \sin 2x_2 + \sin 2x_3 \) is expressed as \( 2(\sin x_1 \cos x_1 + \sin x_2 \cos x_2 + \sin x_3 \cos x_3) \).
• Each component can then be assessed and related back to the given condition \( x_1 + x_2 + x_3 = \pi \).

Through systematic application of trigonometric identities, complex equations simplify, making it easier to prove statements or relate terms.
By leveraging identities like fallen angles, flipped sinusoidal functions, or exclusivity within closed trigonometric equations, solutions can be derived systematically. This approach turns intricate problems into more intuitive forms, ultimately aligning components into the desired equivalence.