Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ 4 \cos x \cos 2 x \sin 3 x=\sin 2 x+\sin 4 x+\sin 6 x $$

Step-by-Step Solution

Answer

The identity holds by rearranging terms using trigonometric identities.

Step 1: Rewrite the Left Side Using Double and Triple Angle Formulas

We start by using trigonometric identities to rewrite the product of cosines and sine on the left side. Recall the triple and double-angle formulas: - For sine: \( \sin 3x = 3 \sin x - 4 \sin^3 x \)- For cosine: \( \cos 2x = 2 \cos^2 x - 1 \) and \( \cos 3x = 4 \cos^3 x - 3 \cos x \).The goal is to simplify \(4 \cos x \cos 2x \sin 3x \) into a form we can manipulate further.

Step 2: Simplify the Expression

Substitute the double-angle formula for \(\cos 2x\) and the expression for \(\sin 3x\):\[4 \cos x (2 \cos^2 x - 1)(3 \sin x - 4 \sin^3 x)\]This expression is complex, so look for opportunities to simplify using basic trigonometric identities: \( \sin^2 x = 1 - \cos^2 x \). The focus will be on parts we can reduce further, especially using known identities to reduce the overall expression.

Step 3: Use Trigonometric Identities

Apply a product-to-sum formula if needed. For example:\(2 \cos a \cos b = \cos(a+b) + \cos(a-b)\)and\( 2 \sin a \cos a = \sin 2a \).Apply these identities to break down the expressions into sums of sines and cosines.

Step 4: Match Terms with the Right Side

The goal is now to split the left side expression so it matches \(\sin 2x + \sin 4x + \sin 6x\). By handling products using any known identity, either as a sum or difference, confirm that each component matches or simplifies directly to the right side.

Step 5: Final Verification

Carefully verify each part: check if terms match, rearranging if necessary. Each trigonometric term from the left (derived expression) should correspond directly to the terms on the right side.

Key Concepts

double-angle formulas

In trigonometry, double-angle formulas are essential tools used to simplify expressions or solve equations involving trigonometric functions. These formulas are derived from the sum formulas and they express trigonometric functions of double angles (like \(2x\)) in terms of single angles. Let's explore some key double-angle formulas that are most often used:

• For cosine: \(\cos 2x = 2\cos^2 x - 1\) or, alternatively, \(\cos 2x = 1 - 2\sin^2 x\)
• For sine: \(\sin 2x = 2\sin x \cos x\)
• For tangent: \(\tan 2x = \frac{2\tan x}{1 - \tan^2 x}\)

In our problem, we use the formula for cosine, \(\cos 2x = 2\cos^2 x - 1\), to simplify parts of our trigonometric expression before further breaking it down using other identities.
Understanding these identities allows us to transition effectively from more complicated expressions to manageable forms.

triple-angle formulas

Triple-angle formulas are the next step beyond double-angle formulas, involving trigonometric expressions where angles are tripled, like \(3x\). These can be particularly useful when needing to express parts of an equation in simpler terms:

• For sine: \(\sin 3x = 3\sin x - 4\sin^3 x\)
• For cosine: \(\cos 3x = 4\cos^3 x - 3\cos x\)

In our example, we make use of the formula for sine, \(\sin 3x = 3\sin x - 4\sin^3 x\), to transform complex products into a form that is easier to manipulate.
By substituting \(\sin 3x\) with this triple-angle identity, we're able to leverage other trigonometric identities to break down and verify the given expression.
These transformations are crucial, especially when working towards showing equivalence between two trigonometric expressions by manipulating and reiterating components to correspond to expected outcomes perfectly.

product-to-sum formulas

Product-to-sum formulas are trigonometric identities that convert products of sines and cosines into sums or differences of these functions. This simplification process is helpful in reducing multiplicative forms into additive forms, which are often easier to handle and compare:

• \(\sin a \sin b = \frac{1}{2} [\cos(a-b) - \cos(a+b)]\)
• \(\cos a \cos b = \frac{1}{2} [\cos(a+b) + \cos(a-b)]\)
• \(\sin a \cos b = \frac{1}{2} [\sin(a+b) + \sin(a-b)]\)

In the context of the exercise, we employ these formulas to transition between product and sum forms. This helps simplify matching terms between the two sides of the identity equation. For instance, converting \(2 \cos a \cos b = \cos(a+b) + \cos(a-b)\) enables us to rearrange and simplify the left-hand side to match predefined trigonometric sums on the right.
Mastery of product-to-sum transformations is incredibly beneficial when verifying identities, as it opens up new routes to achieving desired equation structures efficiently.