Math Problem Statement
give me the solution of this math question
Solution
Let's re-examine the problem and solve it step-by-step.
Given the parametric equations:
The point passes through the x-axis at the origin and three more times. This occurs when .
Step 1: Set to 0
Step 2: Solve for
Using the sum-to-product identities:
This product equals zero if either factor is zero:
Step 3: Solve
Where is any integer. We need to find values of within the given range .
Possible values of :
Step 4: Solve
Where is any integer. We need to find values of within the given range .
Possible values of :
Step 5: Combine and filter the solutions
We discard since we are looking for values where crosses the x-axis besides the origin. We also discard as they are outside the given range.
Thus, the valid values of are:
Final Answer
[ [t_1, t_2, t_3] = \left[ \frac{4}{3} \pi, -\frac{4}{3} \pi, \frac{8}{3} \pi \right] ]
If you have any further questions or need more details, feel free to ask! Here are some questions you might be interested in:
- How can I verify the solutions for ?
- What is the geometric interpretation of this parametric equation?
- How do I apply these parametric equations in a different context?
- Can you provide a graph for better visualization?
- Are there alternative methods to solve this problem?
Tip: When solving trigonometric equations, always consider using identities to simplify the expressions before solving them.
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Trigonometric Identities
Formulas
Sum-to-product identities
Theorems
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Suitable Grade Level
Advanced High School
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