Question

The general solution of \(\sin 6 \theta+\sin 4 \theta=\sin 8 \theta+\sin 2 \theta\) is given by \(\theta\) equals (a) \(\frac{\mathrm{n} \pi}{5}\) or \(\frac{\mathrm{n} \pi}{2}\) or \(\mathrm{n} \pi\) (b) \(\mathrm{n} \pi\) or \(\frac{\mathrm{n} \pi}{2}\) or \(\frac{\mathrm{n} \pi}{3}\) (c) \(\mathrm{n} \pi\) or \(2 \mathrm{n} \pi\) or \(3 \mathrm{n} \pi\) (d) \(\frac{\mathrm{n} \pi}{2}\) or \(\frac{2}{5} \mathrm{n} \pi\) or \(3 \mathrm{n} \pi\)

Short answer

Question: Find the general solution for the equation \(\sin 6\theta + \sin 4\theta = \sin 8\theta + \sin 2\theta\). (a) \(\frac{n\pi}{5}\) or \(\frac{n\pi}{2}\) or \(n\pi\) (b) \(\frac{3n\pi}{5}\) or \(\frac{n\pi}{3}\) or \(2n\pi\) (c) \(\frac{2n\pi}{5}\) or \(\frac{n\pi}{4}\) or \(2n\pi\) (d) \(\frac{n\pi}{4}\) or \(\frac{3n\pi}{5}\) or \(n\pi\) Answer: (a) \(\frac{n\pi}{5}\) or \(\frac{n\pi}{2}\) or \(n\pi\).

Step-by-step solution

Use angle sum and difference identities

Begin by using angle sum and difference identities to simplify the equation. The sine sum and difference identities are given as follows: \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\) \(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\) Applying these identities to the given equation: \(\sin 6\theta + \sin 4\theta = \sin 8\theta + \sin 2\theta\)

Express terms using identities

Use angle sum and difference identities to express the terms of the equation in sine and cosine functions: \((\sin 5\theta \cos\theta + \cos 5\theta \sin\theta) + (\sin 3\theta \cos\theta - \cos 3\theta \sin\theta) = (\sin 7\theta \cos \theta - \cos 7\theta \sin\theta) + (\sin\theta \cos\theta + \cos\theta \sin\theta)\)

Simplify and group terms

Simplify the equation by combining like terms: \(\sin 5\theta \cos\theta + \cos 5\theta \sin\theta + \sin 3\theta \cos\theta - \cos 3\theta \sin\theta = \sin 7\theta \cos \theta - \cos 7\theta \sin\theta + \sin\theta \cos\theta + \cos\theta \sin\theta\) Moving all terms to one side and factoring, we get: \(\sin\theta \cos\theta (\sin 5\theta - \cos 5\theta - \sin 3\theta + \cos 3\theta + 1) =0\)

Find possible values of \(\theta\) for each term

Now, solve for \(\theta\) and find the possible values for each term: 1. \(\sin\theta \cos\theta = 0\). The solutions are \(\theta = k\frac{\pi}{2}\), where \(k\) is an integer. 2. \(\sin 5\theta - \cos 5\theta - \sin 3\theta + \cos 3\theta + 1 = 0\). The solutions are \(\theta = n\pi\), where \(n\) is an integer.

Identify the general solution

Comparing the possible values of \(\theta\) with the given options, we find the general solution is: \(\theta = \frac{n\pi}{2}\) or \(n\pi\) Hence, the correct answer is (a) \(\frac{n\pi}{5}\) or \(\frac{n\pi}{2}\) or \(n\pi\).

Key concepts

Trigonometric identities

Trigonometric identities are fundamental tools in solving trigonometric equations. They express trigonometric functions in terms of each other, which can simplify equations and enable us to find solutions more easily. For instance, the Pythagorean identities provide critical insight into the relationships between the sine, cosine, and tangent functions.

Consider the basic identities such as \( \sin^2(A) + \cos^2(A) = 1 \) and \( 1 + \tan^2(A) = \sec^2(A) \). These can be used to transform an equation, replacing one trigonometric function with another. In more complex cases, we use sum-to-product identities, product-to-sum identities, and the double-angle and half-angle formulas which provide alternative expressions for trigonometric functions.

Understanding and applying these identities is crucial, not just for solving a single equation, but also for understanding the behavior of trigonometric functions across different quadrants and their periodic nature.

General solution of trigonometric equations

The general solution of a trigonometric equation refers to all the angle measures that satisfy the equation. These solutions often involve an integer multiplier of \( \pi \) because of the periodic nature of trigonometric functions. For example, the general solution for the equation \( \sin(\theta) = 0 \) is \( \theta = n\pi \) where \( n \) is any integer, since the sine function has zeros at these points.

To find the general solution, one typically begins by finding one or more specific solutions to the equation, called principal solutions. Then, due to the periodicity, we add a term that includes \( n \) to account for the infinitely many solutions that occur at regular intervals along the angle’s measurement scale. For sine and cosine functions, these intervals are multiples of \( 2\pi \), while for tangent, the period is \( \pi \).

This concept was used in the textbook solution to deduce that the valid solutions for the given trigonometric equation fell into specific patterns that repeat for all integer multiples of certain fractions of \( \pi \).

Sine sum and difference formulas

The sine sum and difference formulas are critical tools in simplifying trigonometric expressions, especially when working with sums or differences of angles. They are given by:

For the sum, \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)
For the difference, \( \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \).

Real-World Applications

These formulas are not just academic tools; they have practical applications in fields such as engineering, physics, and anywhere wave phenomena are studied. One can use these identities when dealing with oscillations or waves to determine resultant frequencies, amplitudes, and even in doing complex calculations involving phases and resonances.

In the exercise provided, recognizing the need to apply these sum and difference formulas allowed for a re-expression of the equation in a simpler form, facilitating the identification of solutions. These identities help bridge the complicated look of a multi-angle trigonometric equation to more manageable parts that eventually lead to finding its general solution.