Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Refer to Exercise 47 of Section 7.4. The graph of the equation \(y=\cos 3 x-3 \cos x\) has 13 turning points for \(-2 \pi \leq x \leq 2 \pi\). The \(x\)-coordinates of these points are solutions of the equation \(\sin 3 x-\sin x=0\). Use a sum-toproduct formula to find these \(x\)-coordinates.

Step-by-Step Solution

Answer

The solutions are the \(x\)-coordinates where \(2\cos(2x)\sin(x)=0\).

Step 1: Understanding the Problem

We need to find the \(x\)-coordinates of the turning points of the function \(y = \cos 3x - 3 \cos x\). The equation that we have to solve is \(\sin 3x - \sin x = 0\). For this, we will use a trigonometric identity to rewrite it in a solvable form.

Step 2: Using the Identity

The sum-to-product formulas in trigonometry can convert sums or differences of sines into products. The identity we will use here is:\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]Apply this identity to \(\sin 3x - \sin x = 0\). Here \(A = 3x\) and \(B = x\), so:\[ \sin 3x - \sin x = 2 \cos\left(\frac{3x + x}{2}\right) \sin\left(\frac{3x - x}{2}\right) = 2 \cos(2x) \sin(x) = 0 \]

Step 3: Solving the Product Equation

The product \(2 \cos(2x) \sin(x) = 0\) implies that either \(\cos(2x) = 0\) or \(\sin(x) = 0\). We will solve both equations separately to find the solutions in the interval \([-2\pi, 2\pi]\).

Step 4: Solve \(\cos(2x) = 0\)

The equation \(\cos(2x) = 0\) has solutions of the form:\[ 2x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z} \]Thus, \(x = \frac{\pi}{4} + \frac{n\pi}{2}\). Calculating for each \(n\) fitting within \(-2\pi \leq x \leq 2\pi\), solve for all \(x\).

Step 5: Solve \(\sin(x) = 0\)

The equation \(\sin(x) = 0\) has solutions of the form:\[ x = n\pi, \quad n \in \mathbb{Z} \]Calculate for each \(n\) such that \(-2\pi \leq x \leq 2\pi\). These are also valid \(x\)-coordinates where the product becomes zero.

Step 6: Combine and Verify Solutions

Combine solutions from \(\cos(2x) = 0\) and \(\sin(x) = 0\). Ensure each found solution is unique and falls within the range \([-2\pi, 2\pi]\). Count the total number of solutions and verify there are 13 distinct points.

Key Concepts

Turning Points

Turning points in the context of trigonometric functions indicate where the graph changes direction - from increasing to decreasing or vice versa. For the equation \(y = \cos(3x) - 3\cos(x)\), these points occur where the derivative \(y'\) is equal to zero or undefined, indicating horizontal tangent lines on the curve. In this problem, the equation \(\sin(3x) - \sin(x) = 0\) is directly related to finding these turning points.

• When we solve this equation, we identify the \(x\)-coordinates where turning points exist.
• This identification is critical to understanding how the function behaves over the given interval.

These turning points also help graph the function more accurately by providing insights into its minima and maxima between critical points.

Sum-to-Product Formulas

Sum-to-product formulas are useful trigonometric identities that transform the sum or difference of two trigonometric functions into a product. This is particularly advantageous in solving equations like \(\sin(A) - \sin(B) = 0\). One such formula is:\[\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)\]Applying this formula helps simplify potentially complex equations, enabling easier solution derivation.

• The transformation allows us to change the problem from dealing with sums to dealing with products.
• Solving the resulting product is often simpler since it can be split into separate equations.

By using these identities, complex trigonometric equations become manageable and solvable.

Trigonometric Equations

Solving trigonometric equations involves finding the angles \(x\) that satisfy given conditions. These equations model periodic behaviors and have solutions in specific intervals. For \(\sin 3x - \sin x = 0\), the solutions occur at particular intervals based on the periodicity of sine and cosine functions.

• The equation was transformed using the sum-to-product formula to \(2 \cos(2x) \sin(x) = 0\).
• The resulting product allows us to split the equation into simpler parts, \(\cos(2x) = 0\) and \(\sin(x) = 0\).

The process involves identifying each solution within a given range, like \(-2\pi \leq x \leq 2\pi\), ensuring all solutions are covered. This often results in multiple solutions due to the cyclical nature of trigonometric functions.

Solving Trigonometric Equations Step by Step

Solving trigonometric equations follows a systematic approach, breaking down complex problems into simpler, manageable steps.**Step 1: Transformation**

• Use identities like sum-to-product to simplify complex expressions.
• For example, convert \(\sin 3x - \sin x = 0\) using an identity to facilitate the solving process.

**Step 2: Solve Resulting Equations**

• Split the resulting product equation, such as \(2 \cos(2x) \sin(x) = 0\), into separate solvable parts.
• Handle each part independently: solve \(\cos(2x) = 0\) and \(\sin(x) = 0\).

**Step 3: Combine Solutions**

• Gather solutions from each part to get the complete set of \(x\)-coordinates.
• Verify all solutions fall within the defined interval and are distinct, leading to the enumeration of solutions such as 13 unique turning points in this exercise.

This structured approach ensures a clear path to correctly solving trigonometric equations, crucial for mastering these mathematical concepts.