What are the general solutions of some trigonometric equations?
Understand the Problem
The question is asking to identify various general solutions to trigonometric equations based on specific conditions related to sine and cosine values.
Answer
- \( \theta = n\pi, n \in \mathbb{Z} \) (for sine zero) - \( \theta = (2n+1)\frac{\pi}{2}, n \in \mathbb{Z} \) (for cosine zero) - Other cases similarly defined as per the listed conditions.
Answer for screen readers
The general solutions for the cases presented are:
(a) ( \theta = n\pi, n \in \mathbb{Z} )
(b) ( \theta = (2n+1)\frac{\pi}{2}, n \in \mathbb{Z} )
(c) ( \theta = n\pi + (-1)^n \frac{\pi}{2}, n \in \mathbb{Z} )
(d) ( \theta = n\pi + (-1)^n \theta, n \in \mathbb{Z} )
(e) ( \theta = 2n\pi \pm \alpha, n \in \mathbb{Z}, \alpha \in [0, \pi] )
(f) ( \theta = n\pi + \frac{\pi}{2}, n \in \mathbb{Z} )
(g) ( \theta = 2n\pi, n \in \mathbb{Z} )
(h) ( \theta = (4n + 1)\frac{\pi}{2}, n \in \mathbb{Z} )
(i) ( \theta = n\pi \pm \alpha, n \in \mathbb{Z} )
(j) ( \theta = n\pi + (-1)^n \frac{\pi}{2}, n \in \mathbb{Z} )
(k) ( \theta = \frac{n\pi}{2} + (-1)^{n-1} \frac{n\pi}{2}, n \text{ (odd integer)} )
Steps to Solve
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Identifying Trigonometric Values
For each case in the provided equations, identify the specific trigonometric values (sine or cosine) that meet the given conditions.
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Applying General Solution Formulas
Utilize the general solution formulas given for sine and cosine functions. For example:
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For ( \sin \theta = 0 ), the general solution is ( \theta = n\pi ) where ( n \in \mathbb{Z} ).
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For ( \cos \theta = 0 ), the general solution is ( \theta = \frac{(2n+1)\pi}{2} ) where ( n \in \mathbb{Z} ).
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Handling the Particular and Principal Solutions
In some cases, derive the principal solution or particular solution from the general solution to meet specific constraints or intervals.
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Consideration of Different Cases
Make sure to analyze and categorize each case based on the ranges or specific angles (for instance, ensuring that angles remain within specific quadrants when necessary).
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Summary of Solutions
Compile all derived general solutions for sine and cosine based on the identified cases, clearly stating the conditions.
The general solutions for the cases presented are:
(a) ( \theta = n\pi, n \in \mathbb{Z} )
(b) ( \theta = (2n+1)\frac{\pi}{2}, n \in \mathbb{Z} )
(c) ( \theta = n\pi + (-1)^n \frac{\pi}{2}, n \in \mathbb{Z} )
(d) ( \theta = n\pi + (-1)^n \theta, n \in \mathbb{Z} )
(e) ( \theta = 2n\pi \pm \alpha, n \in \mathbb{Z}, \alpha \in [0, \pi] )
(f) ( \theta = n\pi + \frac{\pi}{2}, n \in \mathbb{Z} )
(g) ( \theta = 2n\pi, n \in \mathbb{Z} )
(h) ( \theta = (4n + 1)\frac{\pi}{2}, n \in \mathbb{Z} )
(i) ( \theta = n\pi \pm \alpha, n \in \mathbb{Z} )
(j) ( \theta = n\pi + (-1)^n \frac{\pi}{2}, n \in \mathbb{Z} )
(k) ( \theta = \frac{n\pi}{2} + (-1)^{n-1} \frac{n\pi}{2}, n \text{ (odd integer)} )
More Information
These equations describe the general solutions to standard sine and cosine equations, leveraging their periodic nature, which is central to solving trigonometric equations in mathematics. Understanding these solutions aids in solving complex trigonometric problems across various applications in physics and engineering.
Tips
- Mixing up the intervals or domains of the solutions, particularly for trigonometric functions can lead to incorrect evaluations.
- Forgetting the periodicity of sine and cosine functions, which requires you to add multiples of ( \pi ) or ( 2\pi ) appropriately.
- Failing to account for both positive and negative possibilities in the general solutions.
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