Finding All Solutions for the Equation 0 = 2p cos x sin x
To find all solutions to the equation 0 = 2p cos x sin x within the interval [0, 2π], let’s analyze the equation step by step.
Step 1: Simplify the Equation
The original equation can be factored as:
- 0 = 2p cos x sin x
This can be rewritten using the property that the product of two terms is zero if at least one of the terms is zero:
- 2p = 0 or cos x sin x = 0
Step 2: Analyze Each Factor
Factor 1: 2p = 0
This gives us:
- If p = 0, the equation is satisfied for any value of \(x\). Therefore, all x in the interval [0, 2π] are solutions.
Factor 2: cos x sin x = 0
Next, we need to find out when \(cos x\) or \(sin x\) is equal to zero:
- 1. \(cos x = 0\)
- 2. \(sin x = 0\
Finding Solutions for Each
1. Solutions for \(cos x = 0\
Cosine is zero at:
- \(x = \frac{\pi}{2}\)
- \(x = \frac{3\pi}{2}\)
2. Solutions for \(sin x = 0\
Sine is zero at:
- \(x = 0\)
- \(x = \pi\)
- \(x = 2\pi\)
Step 3: Combine All Solutions
Combining all of the obtained solutions, we have:
- From p = 0: \(x \in [0, 2\pi]\) (all values)
- From cos x = 0: \(x = \frac{\pi}{2}, \frac{3\pi}{2}\)
- From sin x = 0: \(x = 0, \pi, 2\pi\)
Final List of Solutions
Therefore, the complete list of solutions in the interval [0, 2π] is:
- 0
- \(\frac{\pi}{2}\)
- \(\pi\)
- \(\frac{3\pi}{2}\)
- 2π
In summary, all solutions are valid values within the range of the given interval, and you can find them either by setting p = 0 or finding the known points where cos x or sin x equals zero.