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Magazine Chapter 10: Problem 57
Solve the equation, giving the exact solutions which lie in \([0,2 \pi)\). $$ \sin (6 x)+\sin (x)=0 $$
Short Answer
Expert verified
The exact solutions are: 0, \(\frac{2\pi}{7}\), \(\frac{4\pi}{7}\), \(\frac{6\pi}{7}\), \(\frac{8\pi}{7}\), \(\frac{10\pi}{7}\), \(\frac{12\pi}{7}\), \(\frac{\pi}{5}\), \(\frac{3\pi}{5}\), \(\pi\), \(\frac{7\pi}{5}\), and \(\frac{9\pi}{5}\).
Step by step solution
01
Use the Sum-to-Product Identities
The equation is \( \sin(6x) + \sin(x) = 0 \). Use the sum-to-product identities: \[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \] Apply it with \( A = 6x \) and \( B = x \), resulting in: \[ 2 \sin\left(\frac{7x}{2}\right) \cos\left(\frac{5x}{2}\right) = 0 \] This equation holds if either the sine term or the cosine term is zero.
02
Solve \( \sin\left(\frac{7x}{2}\right) = 0 \)
Solve \( \sin\left(\frac{7x}{2}\right) = 0 \) using the fact that the sine function is zero at multiples of \( \pi \). So, \( \frac{7x}{2} = n\pi \) for some integer \( n \). Hence, \( x = \frac{2n\pi}{7} \). Find \( x \) in \[ [0, 2\pi) \] which can be done by checking integer values of \( n \) that keep \( x \) less than \( 2\pi \):- For \( n = 0 \), \( x = 0 \).- For \( n = 1 \), \( x = \frac{2\pi}{7} \).- For \( n = 2 \), \( x = \frac{4\pi}{7} \).- For \( n = 3 \), \( x = \frac{6\pi}{7} \).- For \( n = 4 \), \( x = \frac{8\pi}{7} \).- For \( n = 5 \), \( x = \frac{10\pi}{7} \).- For \( n = 6 \), \( x = \frac{12\pi}{7} \).Each of these values satisfies the condition \( x \in [0, 2\pi) \).
03
Solve \( \cos\left(\frac{5x}{2}\right) = 0 \)
Solve \( \cos\left(\frac{5x}{2}\right) = 0 \) using the fact that the cosine function is zero at odd multiples of \( \frac{\pi}{2} \). So, \( \frac{5x}{2} = (2m+1)\frac{\pi}{2} \) for some integer \( m \). Hence, \( x = \frac{(2m+1)\pi}{5} \). Find \( x \) in \[ [0, 2\pi) \] which can be done by checking integer values of \( m \):- For \( m = 0 \), \( x = \frac{\pi}{5} \).- For \( m = 1 \), \( x = \frac{3\pi}{5} \).- For \( m = 2 \), \( x = \frac{5\pi}{5} = \pi \).- For \( m = 3 \), \( x = \frac{7\pi}{5} \).- For \( m = 4 \), \( x = \frac{9\pi}{5} \).These values satisfy the condition \( x \in [0, 2\pi) \).
04
Combine the Solutions
Combine solutions from both Steps 2 and 3. From Step 2, the solutions are: \[ x = 0, \frac{2\pi}{7}, \frac{4\pi}{7}, \frac{6\pi}{7}, \frac{8\pi}{7}, \frac{10\pi}{7}, \frac{12\pi}{7} \].From Step 3, the solutions are:\[ x = \frac{\pi}{5}, \frac{3\pi}{5}, \pi, \frac{7\pi}{5}, \frac{9\pi}{5} \].Make sure no solutions are repeated. These are all the solutions within the interval \( [0, 2\pi) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum-to-Product Identities
The sum-to-product identities are a set of mathematical formulas used to transform the sum of trigonometric functions into a product of trigonometric functions. This is extremely useful for solving trigonometric equations, as it can simplify the process by reducing the complexity of expressions.
- Form: The identity for sine states that \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \).
- Purpose: By transforming sums into products, we can leverage zero-product properties, which make it easier to identify solutions.
Sine Function
The sine function is a fundamental trigonometric function, closely related to the geometry of circles. It represents one of the simplest waveforms in mathematics and is periodic in nature.
- Nature: Sine's primary characteristic is that it outputs values between -1 and 1. It's periodic with a cycle repeating every \( 2\pi \).
- Zeros: The sine function is zero at all integer multiples of \( \pi \). Mathematically, this is expressed as \( \sin(n\pi) = 0 \) for any integer \( n \).
Cosine Function
Much like its counterpart, the cosine function is another core trigonometric function. It also depicts oscillatory motion and has similar properties but phase-shifted relative to the sine.
- Nature: Cosine also has a range from -1 to 1, with the periodicity of \( 2\pi \). It peaks at 1 and -1, synchronized with sine's zero points.
- Zeros: The cosine function becomes zero at odd multiples of \( \frac{\pi}{2} \), meaning \( \cos\left((2m+1)\frac{\pi}{2}\right) = 0 \).
Exact Solutions
Finding exact solutions in trigonometry involves solving equations where both inputs and outputs maintain their mathematical integrity鈥攐ften resulting in expressions involving \( \pi \).
- Interval Constraint: Solutions must fit within a specified interval, often \( [0, 2\pi) \), ensuring meaningful, comparable results.
- Strategic Solving: By addressing the transformed product identity as separate zero-point problems (i.e., \( \sin\left(\frac{7x}{2}\right) = 0 \) and \( \cos\left(\frac{5x}{2}\right) = 0 \)), each variable specific to its cycle is located.
