Problem 63
Question
\(61-66\) Write the sum as a product. $$ \cos 4 x-\cos 6 x $$
Step-by-Step Solution
Verified Answer
\(2 \sin(5x) \sin(x)\)
1Step 1: Identify the Trigonometric Identity
To write the difference of cosines as a product, we use the trigonometric identity:\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]Here, \(A = 4x\) and \(B = 6x\).
2Step 2: Calculate \(\frac{A+B}{2}\)
Substitute \(A = 4x\) and \(B = 6x\) into \(\frac{A+B}{2}\):\[ \frac{4x + 6x}{2} = \frac{10x}{2} = 5x \]
3Step 3: Calculate \(\frac{A-B}{2}\)
Substitute \(A = 4x\) and \(B = 6x\) into \(\frac{A-B}{2}\):\[ \frac{4x - 6x}{2} = \frac{-2x}{2} = -x \]
4Step 4: Apply the Identity
Using the identity from Step 1, substitute the calculated expressions from Steps 2 and 3:\[ \cos 4x - \cos 6x = -2 \sin(5x) \sin(-x) \]
5Step 5: Simplify the Expression
Since \(\sin(-x) = -\sin(x)\), substitute to simplify:\[ -2 \sin(5x) \sin(-x) = -2 \sin(5x) (-\sin(x)) = 2 \sin(5x) \sin(x) \]
Key Concepts
Difference of CosinesProduct-to-Sum FormulasSimplifying Trigonometric Expressions
Difference of Cosines
The difference of cosines is an important trigonometric expression that can be simplified using specific identities. In particular, when you need to express the difference of two cosine functions, such as \( \cos 4x - \cos 6x \), you can utilize a trigonometric identity to transform this difference into a product.
The formula used for this transformation is:
\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
This identity allows us to rewrite a difference of cosines as a product of sines, which can simplify further calculation or integration. By replacing \(A\) with \(4x\) and \(B\) with \(6x\), you can calculate \(\frac{A+B}{2}\) and \(\frac{A-B}{2}\) to use in the formula.
Applying the identity transforms the expression into \(-2 \sin(5x) \sin(-x)\), which can further be simplified. This step is crucial when dealing with complex trigonometric expressions.
The formula used for this transformation is:
\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
This identity allows us to rewrite a difference of cosines as a product of sines, which can simplify further calculation or integration. By replacing \(A\) with \(4x\) and \(B\) with \(6x\), you can calculate \(\frac{A+B}{2}\) and \(\frac{A-B}{2}\) to use in the formula.
Applying the identity transforms the expression into \(-2 \sin(5x) \sin(-x)\), which can further be simplified. This step is crucial when dealing with complex trigonometric expressions.
Product-to-Sum Formulas
Product-to-sum formulas are indispensable tools in trigonometry that help convert products of trigonometric functions into sums or differences. These formulas are especially helpful in simplifying expressions or solving equations.
While the specific exercise discussed is not directly about a product-to-sum conversion, the identity \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \) used here resembles this strategy because it transforms a difference into a product. This type of manipulation is beneficial in calculus and physics where solving and integrating trigonometric functions is common.
Understanding how to move between products and sums, or vice versa, using these identities can enhance problem-solving skills in trigonometry, making it easier to deal with integrals and other complex expressions.
While the specific exercise discussed is not directly about a product-to-sum conversion, the identity \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \) used here resembles this strategy because it transforms a difference into a product. This type of manipulation is beneficial in calculus and physics where solving and integrating trigonometric functions is common.
Understanding how to move between products and sums, or vice versa, using these identities can enhance problem-solving skills in trigonometry, making it easier to deal with integrals and other complex expressions.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions often involves applying known identities and algebraic manipulation to express complex trigonometric statements more simply. In the given example, one notable simplification occurs after applying the difference of cosines identity.
Initially, the expression \(-2 \sin(5x) \sin(-x)\) results from applying the identity to \(\cos 4x - \cos 6x\). However, it is essential to recognize that \(\sin(-x) = -\sin(x)\), which leads to further simplification:
\[ -2 \sin(5x) \sin(-x) = -2 \sin(5x) (-\sin(x)) = 2 \sin(5x) \sin(x) \]
This step is crucial as it ensures that the expression is in its simplest form, making further calculations and interpretations easier.
Simplifying trigonometric expressions can reduce unnecessary complexity, making it simpler to evaluate or integrate the expressions, which is often required in advanced mathematics courses.
Initially, the expression \(-2 \sin(5x) \sin(-x)\) results from applying the identity to \(\cos 4x - \cos 6x\). However, it is essential to recognize that \(\sin(-x) = -\sin(x)\), which leads to further simplification:
\[ -2 \sin(5x) \sin(-x) = -2 \sin(5x) (-\sin(x)) = 2 \sin(5x) \sin(x) \]
This step is crucial as it ensures that the expression is in its simplest form, making further calculations and interpretations easier.
Simplifying trigonometric expressions can reduce unnecessary complexity, making it simpler to evaluate or integrate the expressions, which is often required in advanced mathematics courses.
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