Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Use sum-to-product formulas to find the solutions of the equation. $$ \cos x=\cos 3 x $$

Step-by-Step Solution

Answer

The solutions are \( x = n\pi/2 \) where \( n \) is an integer.

Step 1: Apply the Sum-to-Product Formula for Cosine

The equation is given as \( \cos x = \cos 3x \). The sum-to-product formula for cosine is: \[ \cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \]We set \( A = 3x \) and \( B = x \). Thus, \( \cos 3x - \cos x = 0 \). This gives us:\[ -2 \sin \left( \frac{4x}{2} \right) \sin \left( \frac{2x}{2} \right) = 0 \]Simplifying, we have:\[ -2 \sin (2x) \sin (x) = 0 \]

Step 2: Solve the Equation by Setting Each Factor to Zero

To solve the equation \(-2 \sin(2x) \sin(x) = 0\), we set each factor to zero:1. \( \sin(2x) = 0 \)2. \( \sin(x) = 0 \)We will solve these separately to find \(x\).

Step 3: Solve \(\sin(2x) = 0\)

The solutions to \( \sin(2x) = 0 \) are given by:\[ 2x = n\pi \quad \Rightarrow \quad x = \frac{n\pi}{2} \]where \(n\) is an integer.

Step 4: Solve \(\sin(x) = 0\)

The solutions to \( \sin(x) = 0 \) are given by:\[ x = m\pi \]where \(m\) is an integer.

Step 5: Combine Solutions

Both sets of solutions are for \(x\), so we combine them:\[ x = n\pi/2 \quad \text{and} \quad x = m\pi \]Here, \( n \) could be even or odd, while \( m \) is any integer. Since all solutions from \( x = m\pi \) are included in \( x = n\pi/2 \) when \( n \) is even, the complete solution is given by the combinations of these. For simplicity, candidates such as \( x = 0, \pi/2, \pi, 3\pi/2, \ldots \) cover all cases involving both conditions.

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions that involve trigonometric functions and are set equal to each other or to a constant. These types of equations are fundamental in trigonometry because they allow us to find unknown angles or lengths in geometric problems. In the given exercise, we encounter a trigonometric equation: \( \cos x = \cos 3x \). Solving trigonometric equations often involves manipulating the equation using identities, transforming it into a simpler form that can be solved more easily.

• Trigonometric equations can have multiple solutions due to the periodic nature of trigonometric functions.
• It is essential to consider all possible solutions within a given interval.
• Advanced techniques, such as the sum-to-product identities, help simplify and solve these equations efficiently.

In essence, solving trigonometric equations requires a solid understanding of trigonometric identities and an ability to recognize patterns within the functions.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, commonly denoted as \( \cos \). It is defined as the adjacent side over the hypotenuse in a right-angled triangle. Cosine is a periodic function, which means it repeats its values at regular intervals. The standard period of \( \cos x \) is \( 2\pi \).

• The cosine function is an even function, meaning \( \cos(-x) = \cos(x) \).
• It ranges from -1 to 1, and its graph is a wave that oscillates above and below the x-axis.

In the given exercise, we have to equate two cosine expressions \( \cos x \) and \( \cos 3x \). Utilizing identities can help us solve equations that involve transformations like \( 3x \), allowing us to isolate the variable \( x \) effectively.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are essential tools for simplifying trigonometric expressions and solving equations. In this exercise, we use the sum-to-product identities, particularly:\[\cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right)\]This identity transforms a difference of two cosine functions into a product of sine functions, allowing us to solve the equation step-by-step by breaking it down into more manageable components.

• The sum-to-product identities are valuable for simplifying expressions with multiple trigonometric terms.
• They convert sums or differences into products, which can then be set to zero, making it easier to find solutions.

Using such identities requires familiarity and practice, but they provide a powerful toolset for solving complex trigonometric equations efficiently.

Sin Function

The sine function, represented as \( \sin \), is another cornerstone in trigonometry similar to the cosine function. For an angle in a right-angled triangle, it is defined as the ratio of the opposite side to the hypotenuse. Sine, like cosine, is a periodic function with a period of \( 2\pi \). It also oscillates between -1 and 1.

• The sine function is an odd function, which means \( \sin(-x) = -\sin(x) \).
• The zeros of the sine function, which occur at multiples of \( \pi \), are vital when solving equations like \( \sin(x) = 0 \).

In our exercise, the sine function plays a pivotal role after applying the sum-to-product formulas. We derive two factors: \( \sin(2x) = 0 \) and \( \sin(x) = 0 \). Solving these leads us to the solutions of the original equation. Recognizing where sine equals zero is fundamental in finding all possible solutions efficiently.