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Magazine Chapter 3: Problem 21
Solve: \(\sin 7 x+\sin 4 x+\sin x=0,0 \leq x \leq \frac{\pi}{2}\)
Short Answer
Expert verified
The solutions to the given trigonometric equation within the specified range are \(x=\frac{\pi}{9}\) and \(x=\frac{5\pi}{9}\).
Step by step solution
01
Apply Trigonometric Identities
Express the equation using the sum-to-product identities. These identities are \(\sin a + \sin b = 2 \sin \frac{1}{2}(a+b) \cos \frac{1}{2}(a-b)\) and \(\sin a - \sin b = 2 \cos \frac{1}{2}(a+b) \sin \frac{1}{2}(a-b)\). Apply these to the given problem to get, \(\sin 7x + \sin x = \sin 4x\). Applying sum-to-product identity, this simplifies to \(2 \sin 4x \cos 3x = \sin 4x\).
02
Simplify equation
The equation simplifies further by dividing both sides by \(\sin 4x\), we get \(2 \cos 3x = 1\).
03
Solve for x within the given range
Now solve \(2 \cos 3x = 1\), we have \(\cos 3x = \frac{1}{2}\). Since \(0 \leq x \leq \frac{\pi}{2}\), \(0 \leq 3x \leq \frac{3\pi}{2}\). Now, Cosine function gives \(\frac{1}{2}\) at \(x=\frac{\pi}{3}\) and \(x=\frac{5\pi}{3}\) in the interval \([0, 2\pi]\). Therefore, the solutions are \(x=\frac{\pi}{9}\) and \(x=\frac{5\pi}{9}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum-to-Product Identities
In trigonometry, sum-to-product identities are crucial for transforming the sum or difference of sines or cosines into a product, simplifying complex equations. This can make handling and solving trigonometric equations much easier. Here are the two key identities:
- For sine: \( \sin a + \sin b = 2 \sin \left( \frac{1}{2}(a+b) \right) \cos \left( \frac{1}{2}(a-b) \right) \)
- For sine differences: \( \sin a - \sin b = 2 \cos \left( \frac{1}{2}(a+b) \right) \sin \left( \frac{1}{2}(a-b) \right) \)
Trigonometric Equations
Trigonometric equations involve trigonometric functions, which are solved for specific angles within a given interval. These equations are common in calculus and physics due to their periodic nature. When solving such equations, it's important to:
- Simplify the equation using known identities.
- Find general solutions before narrowing to specific intervals.
Cosine Function
The cosine function, denoted as \( \cos \), is a periodic function with a cycle of \(2\pi\), possessing values from -1 to 1. In this exercise, understanding the behavior of the cosine function is crucial, as it assists in solving the equation \(2 \cos 3x = 1\). The aim was to reach the point where we can easily solve for \(x\).
- Cosine equals \( \frac{1}{2} \) at angles \(x=\frac{\pi}{3}\) and \(x=\frac{5\pi}{3}\) within one cycle \([0, 2\pi]\).
- These roots help in determining specific solutions within given bounds.
Trigonometric Solutions
Finding solutions to trigonometric problems involves identifying angles where a function reaches specific values. After simplifying the problem with identities, the next goal is to solve equations like \(\cos 3x = \frac{1}{2}\). Success requires:
- Recognizing key values where trigonometric functions achieve given outputs.
- Utilizing properties and periodic behaviors effectively.
Interval Analysis
Given a set boundary for solutions, interval analysis simplifies the identification of valid answers within a range. For example, finding solutions within interval \([0, \frac{\pi}{2}]\) requires extra care to ensure all solutions meet interval restrictions.
- Verify each potential solution fits within the specific bounds.
- Consider the effect of trigonometric function periodicity on intervals.
