Trigonometric Identities Exploring 2cos(3θ) And Sin(3θ)cos³(θ) + Cos(3θ)sin³(θ)

In this article, we will delve into two intriguing trigonometric problems. First, we'll tackle the problem where given $2 \\cos \\theta = x + \\frac{1}{x}$, we need to find the value of $2 \\cos 3\\theta$. Then, we'll explore the expression $\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta$ and simplify it using trigonometric identities. These problems not only test our understanding of trigonometric formulas but also our ability to manipulate and apply them effectively. By breaking down each problem step by step, we'll gain a deeper appreciation for the elegance and power of trigonometry.

1. Finding $2 \\cos 3\\theta$ when $2 \\cos \\theta = x + \\frac{1}{x}$

Understanding the Problem

The core of trigonometric identities lies in expressing trigonometric functions in different forms. This question challenges us to find an expression for $2 \\cos 3\\theta$ given that $2 \\cos \\theta = x + \\frac{1}{x}$. To solve this, we will utilize the triple angle formula for cosine, which states that $\\cos 3\\theta = 4\\cos^3 \\theta - 3\\cos \\theta$. By manipulating this formula and using the given equation, we can express $2 \\cos 3\\theta$ in terms of $x$. This problem emphasizes the importance of recognizing and applying the appropriate trigonometric identities to simplify complex expressions. Understanding the interplay between different trigonometric functions allows us to solve problems efficiently and accurately.

Step-by-Step Solution

1. Recall the triple angle formula for cosine: The triple angle formula for cosine is a fundamental identity that relates $\\cos 3\\theta$ to $\\cos \\theta$. It is given by:

   $$\\cos 3\\theta = 4\\cos^3 \\theta - 3\\cos \\theta$$

   This formula is crucial for solving the problem, as it directly links the expression we want to find ($2 \\cos 3\\theta$) to $\\cos \\theta$, which is related to $x$ in the given equation. Memorizing and understanding this identity is key to tackling similar trigonometric problems.

2. Multiply both sides by 2: To find $2 \\cos 3\\theta$, we multiply the entire equation by 2:

   $$2\\cos 3\\theta = 2(4\\cos^3 \\theta - 3\\cos \\theta) = 8\\cos^3 \\theta - 6\\cos \\theta$$

   This step sets up the equation in a form that directly corresponds to the desired expression. By isolating $2 \\cos 3\\theta$, we prepare to substitute the given expression in terms of $x$.

3. Use the given equation: We are given that $2\\cos \\theta = x + \\frac{1}{x}$. Thus, $\\cos \\theta = \\frac{1}{2}\\left(x + \\frac{1}{x}\\right)$.

   This substitution is the bridge that connects the triple angle formula to the variable $x$. By expressing $\\cos \\theta$ in terms of $x$, we can rewrite the entire equation in terms of $x$, leading us closer to the final solution.

4. Substitute $\\cos \\theta$ into the equation for $2\\cos 3\\theta$:

   $$2\\cos 3\\theta = 8\\left[\\frac{1}{2}\\left(x + \\frac{1}{x}\\right)\\right]^3 - 6\\left[\\frac{1}{2}\\left(x + \\frac{1}{x}\\right)\\right]$$

   This substitution replaces $\\cos \\theta$ in the equation with its equivalent expression in terms of $x$. This step is crucial for expressing $2 \\cos 3\\theta$ solely in terms of $x$, which is the goal of the problem.

5. Simplify the expression: Now we simplify the expression step by step:

   $$2\\cos 3\\theta = 8\\left[\\frac{1}{8}\\left(x + \\frac{1}{x}\\right)^3\\right] - 3\\left(x + \\frac{1}{x}\\right)$$

   $$2\\cos 3\\theta = \\left(x + \\frac{1}{x}\\right)^3 - 3\\left(x + \\frac{1}{x}\\right)$$

   This simplification involves basic algebraic manipulations. By carefully expanding and reducing the terms, we can move closer to a more concise expression for $2 \\cos 3\\theta$.

6. Expand $\\left(x + \\frac{1}{x}\\right)^3$:

   $$\\left(x + \\frac{1}{x}\\right)^3 = x^3 + 3x^2\\left(\\frac{1}{x}\\right) + 3x\\left(\\frac{1}{x}\\right)^2 + \\frac{1}{x^3} = x^3 + 3x + \\frac{3}{x} + \\frac{1}{x^3}$$

   Expanding the cubic term is a crucial step. It allows us to break down the expression into individual terms that can be further simplified. The binomial expansion formula is essential here.

7. Substitute the expanded form back into the equation:

   $$2\\cos 3\\theta = \\left(x^3 + 3x + \\frac{3}{x} + \\frac{1}{x^3}\\right) - 3\\left(x + \\frac{1}{x}\\right)$$

   This substitution brings the expanded form back into the main equation, allowing us to combine like terms and simplify the expression further. It's a strategic move to consolidate our progress.

8. Simplify further:

   $$2\\cos 3\\theta = x^3 + 3x + \\frac{3}{x} + \\frac{1}{x^3} - 3x - \\frac{3}{x}$$

   $$2\\cos 3\\theta = x^3 + \\frac{1}{x^3}$$

   The final simplification involves canceling out terms and arriving at the concise form $x^3 + \\frac{1}{x^3}$. This result beautifully expresses $2 \\cos 3\\theta$ in terms of $x$.

Conclusion for Part 1

Therefore, if $2\\cos \\theta = x + \\frac{1}{x}$, then $2\\cos 3\\theta = x^3 + \\frac{1}{x^3}$. This solution highlights the power of trigonometric identities and algebraic manipulation in simplifying complex expressions. The correct answer is option 2) $x^3 + \\frac{1}{x^3}$.

2. Simplifying $\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta$

Understanding the Problem

This problem requires us to simplify the expression $\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta$. To do this, we will need to use trigonometric identities, specifically the triple angle formulas for sine and cosine, and then factor and simplify the resulting expression. The key here is to recognize common factors and apply the identities strategically to reduce the expression to its simplest form. This type of problem tests our ability to not only recall and apply trigonometric identities but also to manipulate algebraic expressions effectively.

Step-by-Step Solution

1. Factor out common terms: The given expression is $\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta$. We can factor out $\\sin \\theta \\cos \\theta$ from both terms:

   $$\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta = \\sin \\theta \\cos \\theta(\\sin^2 \\theta \\cos 3\\theta + \\cos^2 \\theta \\sin 3\\theta)$$

   This factorization is a crucial step that simplifies the expression by isolating common terms. It sets the stage for applying trigonometric identities to the remaining factors.

2. Use the triple angle formulas: Recall the triple angle formulas for sine and cosine:

   $$\\sin 3\\theta = 3\\sin \\theta - 4\\sin^3 \\theta$$

   $$\\cos 3\\theta = 4\\cos^3 \\theta - 3\\cos \\theta$$

   These formulas are essential for breaking down $\\sin 3\\theta$ and $\\cos 3\\theta$ into expressions involving $\\sin \\theta$ and $\\cos \\theta$. This allows us to work towards simplifying the entire expression.

3. Substitute the triple angle formulas: Substitute the triple angle formulas into the factored expression:

   $$\\sin \\theta \\cos \\theta [\\sin^2 \\theta (4\\cos^3 \\theta - 3\\cos \\theta) + \\cos^2 \\theta (3\\sin \\theta - 4\\sin^3 \\theta)]$$

   This substitution is a key step in transforming the expression into a form that can be further simplified. By replacing $\\sin 3\\theta$ and $\\cos 3\\theta$ with their respective expansions, we introduce terms that can be combined and reduced.

4. Expand the terms: Expand the expression inside the brackets:

   $$\\sin \\theta \\cos \\theta [4\\sin^2 \\theta \\cos^3 \\theta - 3\\sin^2 \\theta \\cos \\theta + 3\\cos^2 \\theta \\sin \\theta - 4\\cos^2 \\theta \\sin^3 \\theta]$$

   Expanding the terms allows us to see the individual components of the expression more clearly. This is a crucial step in identifying potential cancellations and simplifications.

5. Rearrange the terms: Rearrange the terms to group similar expressions together:

   $$\\sin \\theta \\cos \\theta [4\\sin^2 \\theta \\cos^3 \\theta - 4\\cos^2 \\theta \\sin^3 \\theta - 3\\sin^2 \\theta \\cos \\theta + 3\\cos^2 \\theta \\sin \\theta]$$

   Rearranging the terms makes it easier to identify common factors and apply further simplifications. Grouping similar terms together is a strategic approach to solving complex expressions.

6. Factor out common factors: Factor out $4\\sin^2 \\theta \\cos^2 \\theta$ from the first two terms and $3\\sin \\theta \\cos \\theta$ from the last two terms:

   $$\\sin \\theta \\cos \\theta [4\\sin^2 \\theta \\cos^2 \\theta(\\cos \\theta - \\sin \\theta) + 3\\sin \\theta \\cos \\theta(-\\sin \\theta + \\cos \\theta)]$$

   This factorization step further simplifies the expression by extracting common factors. It reveals a common binomial factor that can be factored out in the next step.

7. Factor out $(\\cos \\theta - \\sin \\theta)$:

   $$\\sin \\theta \\cos \\theta [(\\cos \\theta - \\sin \\theta)(4\\sin^2 \\theta \\cos^2 \\theta + 3\\sin \\theta \\cos \\theta)]$$

   Factoring out the common binomial $(\\cos \\theta - \\sin \\theta)$ is a significant step that simplifies the expression considerably. It consolidates the terms and sets up the final simplification.

8. Factor out $\\sin \\theta \\cos \\theta$ again:

   $$\\sin \\theta \\cos \\theta [(\\cos \\theta - \\sin \\theta)\\sin \\theta \\cos \\theta(4\\sin \\theta \\cos \\theta + 3)]$$

   This step further simplifies the expression by extracting another common factor, $\\sin \\theta \\cos \\theta$. It prepares the expression for the final simplification using the double angle formula for sine.

9. Simplify the expression:

   $$\\sin^2 \\theta \\cos^2 \\theta (\\cos \\theta - \\sin \\theta)(4\\sin \\theta \\cos \\theta + 3)$$

   Now, use the identity $2\\sin \\theta \\cos \\theta = \\sin 2\\theta$, so $\\sin \\theta \\cos \\theta = \\frac{1}{2}\\sin 2\\theta$:

   $$\\left(\\frac{1}{2}\\sin 2\\theta\\right)^2 (\\cos \\theta - \\sin \\theta)(4\\left(\\frac{1}{2}\\sin 2\\theta\\right) + 3)$$

   $$\\frac{1}{4}\\sin^2 2\\theta (\\cos \\theta - \\sin \\theta)(2\\sin 2\\theta + 3)$$

   The final simplified expression is $\\frac{1}{4}\\sin^2 2\\theta (\\cos \\theta - \\sin \\theta)(2\\sin 2\\theta + 3)$.

Conclusion for Part 2

Therefore, the simplified form of $\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta$ is $\\frac{1}{4}\\sin^2 2\\theta (\\cos \\theta - \\sin \\theta)(2\\sin 2\\theta + 3)$. This problem demonstrates the importance of strategic factorization and the application of trigonometric identities to simplify complex expressions.

These two problems illustrate the beauty and complexity of trigonometric identities. By mastering these identities and practicing algebraic manipulation, we can solve a wide range of trigonometric problems. The key is to break down complex problems into smaller, manageable steps, and to strategically apply the appropriate identities to simplify expressions. Whether it's finding $2\\cos 3\\theta$ or simplifying $\\sin 3\\theta \\cos^3 \\theta + \\cos 3\\theta \\sin^3 \\theta$, a solid understanding of trigonometric principles is essential.