Problem 54 - Algebra and Trigonometry

Question

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\cos 5 \theta-\cos 7 \theta=0\)

Step-by-Step Solution

Answer

\( \theta = \frac{n\pi}{6} \) or \( \theta = m\pi \)

Step 1: Rewrite Using the Sum-to-Product Formula

The Sum-to-Product formula for cosines states that \( \cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \). Here, let \( A = 5 \theta \) and \( B = 7 \theta \). Substituting these values, the equation becomes: \( -2 \sin \left( \frac{5\theta + 7\theta}{2} \right) \sin \left( \frac{5\theta - 7\theta}{2} \right) = 0 \).

Step 2: Simplify and Solve Each Factor

The equation is now \( -2 \sin (6\theta) \sin (-\theta) = 0 \). Since it's a product equal to zero, solve each factor separately: \( \sin (6\theta) = 0 \) and \( \sin (-\theta) = 0 \).

Step 3: Solve \( \sin (6\theta) = 0 \)

The sine function equals zero at \( n\pi \), where \( n \) is an integer. Thus, \( 6\theta = n\pi \). Solving for \( \theta \), we get \( \theta = \frac{n\pi}{6} \).

Step 4: Solve \( \sin (-\theta) = 0 \)

Since \( \sin(-\theta) = -\sin(\theta) \), this implies \( \sin(\theta) = 0 \) as the sine of \( 0 \) and any multiple of \( \pi \) is zero. Thus, \( \theta = m\pi \), where \( m \) is an integer.

Step 5: Conclusion of Solutions

Combining both solutions, \( \theta \) can be \( \frac{n\pi}{6} \) or \( m\pi \). These solutions represent where the original expression is zero, indicating values for \( \theta \) are multiples of \( \pi \) or a sixth of \( \pi \).

Key Concepts

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for any angle. They are essential tools in simplifying complex trigonometric equations and expressions. One key identity used here is the Sum-to-Product formula, which is particularly useful in transforming the difference or sum of trig functions into a product.

The Sum-to-Product formula for cosine is:

• \( \cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \)

This identity transforms a difference between two cosines into a product of sines. It leverages the symmetry and periodic behavior of sinusoidal functions to simplify complex expressions, which is a common technique when solving trigonometric equations.

By applying this identity, we make it much easier to factor and solve trigonometric equations, as was done in the given exercise.

Solving trigonometric equations

Solving trigonometric equations involves finding all angle values that satisfy the equation. These solutions often include a family of angles due to the periodic nature of trigonometric functions.

In the example equation \( \cos 5 \theta - \cos 7 \theta = 0 \), after applying the Sum-to-Product formula, we obtain:

• \( -2 \sin(6\theta) \sin(-\theta) = 0 \)

When dealing with such products set to zero, we apply the zero-product property, which tells us that if a product is zero, at least one of the factors must be zero.

This leads us to solve two simpler equations: \( \sin(6\theta) = 0 \) and \( \sin(-\theta) = 0 \). Each of these can be resolved by considering where the sine function is zero, ultimately leading to solutions based on multiples of \( \pi \). It’s crucial in these problems to remember to check general solutions for radians or degrees to cover all possibilities in the angle period.

Sine function

The sine function is a fundamental trigonometric function with a range of [-1, 1] and is periodic with a period of \( 2\pi \). It is positive in the first and second quadrants and negative in the third and fourth quadrants of the circle.

• Key points of interest are where \( \sin(\theta) = 0 \), which occurs at integer multiples of \( \pi \) (e.g., \( 0, \pi, 2\pi, 3\pi, \ldots \)).

This property was utilized in the exercise solution when solving \( \sin(6\theta) = 0 \) and \( \sin(-\theta) = 0 \). Recognizing these zero-crossings is crucial because they determine the critical angles where the sine factor vanishes.

When solving \( \sin(-\theta) = 0 \), knowing that \( \sin(-\theta) = -\sin(\theta) \) confirms that sine’s zero points remain unchanged, offering a regular pattern of solutions. This contributes to the breadth of the solutions for \( \theta \).

Cosine function

The cosine function, like the sine function, is a core part of trigonometry with a range of [-1, 1] and a period of \( 2\pi \). It describes the horizontal coordinate of a point on the unit circle for a given angle.

The cosine function is important when dealing with equations involving cosines, such as \( \cos 5\theta - \cos 7\theta = 0 \). Initially expressed in terms of differences, the Sum-to-Product formula allows us to transition from cosine differences to sine products.

Cosine also has key landmarks where \( \cos(\theta) = 0 \), occurring at angles like \( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \). However, in the given exercise, the focus transitions from the cosine expression to its sine-equivalent form thanks to these trigonometric identities, highlighting the dynamic interplay between sine and cosine functions.