Problem 127 - Precalculus with Limits

Question

In Exercises 125 - 128, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. \( \left(\cos 4x - \cos 2x\right) / \left(2 \sin 3x\right) = -\sin x \)

Step-by-Step Solution

Answer

The given equation \((\cos 4x - \cos 2x) / (2 \sin 3x) = -\sin x\) is not an identity as the left side simplifies to \(\sin x\) which is not equal to the right side \(-\sin x\). This is confirmed by both graphical and algebraic verification.

Step 1: Graphical Verification

Graph both \((\cos 4x - \cos 2x) / (2 \sin 3x)\) and \(-\sin x\) using a graphing utility. If the two graphs are identical, this verifies the identity graphically.

Step 2: Algebraic Verification - simplify the left side

Use trigonometric identities to simplify the left side of the equation: \((\cos 4x - \cos 2x) / (2 \sin 3x)\). By using the formula for the difference of two cosines, \(cos a - cos b = -2sin(\frac{a+b}{2})sin(\frac{a-b}{2})\), simplify \(\cos 4x - \cos 2x\) and get \(2sin(3x)\sin(x)\). Then divide by \(2 \sin 3x\) which simplifies the left side to \(\sin x\).

Step 3: Comparison of the two sides

Now that the left side of the equation has been simplified to \(\sin x\), compare with the right side of the equation. The right side of the equation is \(-\sin x\). This shows that the given equation is not correct, since \(\sin x\) is not equal to \(-\sin x\). Therefore, the given equation is not an identity.

Key Concepts

Difference of Cosines Formula

The difference of cosines formula is a trigonometric identity that is very useful in simplifying expressions involving cosine functions. This formula states:

• \(\cos a - \cos b = -2 \sin\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right)\)

This transformation helps simplify complicated expressions by breaking them down into products of sines, which can sometimes be easier to manipulate.
In our exercise, the term \(\cos 4x - \cos 2x\) is transformed using this formula. Plugging into the formula, we let \(a = 4x\) and \(b = 2x\), giving us:

• \(-2 \sin\left(\frac{4x + 2x}{2}\right) \sin\left(\frac{4x - 2x}{2}\right)\ = -2 \sin(3x) \sin(x)\)

This results in a simpler expression that can further be processed algebraically. Using such identities is key in tackling complex trigonometric problems and converting them into manageable expressions is a strategic way to verify identities.

Graphical Verification in Trigonometry

Graphical verification involves using technology, like graphing calculators or software, to visually confirm the correctness of a trigonometric identity. The idea is that if both sides of a trigonometric identity produce the same graph, the identity is verified visually.
For this particular exercise, the task is to graph both \(\frac{\cos 4x - \cos 2x}{2 \sin 3x}\) and \(-\sin x\). The graphical comparison should show that the two curve overlays each other across the interval considered.

• If the graphs trace the same path, this suggests they are equivalent and verifies the identity.
• If there are discrepancies in the graphs, the analytic simplification or assumptions made in the original derivations should be reconsidered.

It's important to be precise and use proper settings in graphing tools for an accurate comparison, as minor graphing errors could lead to incorrect conclusions.

Algebraic Verification

Algebraic verification is an analytical method used to confirm that two expressions are identical by simplifying one or both sides using known algebraic or trigonometric identities.
In our problem, algebraic verification was used after applying the difference of cosines formula to transform and simplify the given expression. The left side of the identity was simplified from \(\cos 4x - \cos 2x\) to \(2 \sin 3x \sin x\), and then divided by \(2 \sin 3x\), thus simplifying to \(\sin x\).

• After simplification, the left-hand side was \(\sin x\).
• The right-hand side was \(-\sin x\).

However, these two simplified expressions are not the same, indicating that the original equation given in the exercise is not actually an identity. This points to the critical importance of carefully performing simplifications and comparisons to ensure accuracy in determining identities.

Trigonometric Simplification Techniques

Trigonometric simplification techniques are fundamental tools that help solve complex problems involving trigonometric functions by transforming intricate expressions into simpler forms. These techniques often involve:

• Using well-known identities, such as the Pythagorean identities and angle addition formulas.
• Simplifying using product-to-sum and sum-to-product identities, like the difference of cosines formula.

In our exercise, the simplification was approached by using both the difference of cosines formula and dividing by another trigonometric expression. This brought a complex expression to a much simpler form,\(\sin x\).
It is crucial to

• systematically apply identities and transformations,
• while checking each step to ensure the logical consistency side by side,
• and re-evaluating if an unexpected result emerges, as was seen when the left-hand and right-hand sides did not match.

Simplification techniques require practice to master, but they are an invaluable asset in both verifying and solving trigonometric identities.