Math Problem Statement
Find the zeroes of 2 cot z z
Solution
The function you're working with is , and you are looking for the zeros of this function.
To find the zeros of , we must find values of such that:
Since this is a product, the function will be zero if either factor is zero:
1. Solving
This equation gives us the solution:
2. Solving
The cotangent function is zero whenever for integers . These are the points where the sine function is 1 or -1 (since ).
Thus, the solutions are:
Conclusion
The zeros of are:
Would you like further details on the solution?
Here are 5 related questions:
- What are the properties of the cotangent function?
- How would the graph of look near the zeros?
- What happens to at the points where is undefined?
- Can this problem be extended to complex numbers ?
- How do the zeros of relate to the zeros of ?
Tip: Always check the domain of trigonometric functions, as they can introduce undefined points in the function.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Complex Analysis
Roots of Functions
Formulas
z^2 cot(z) = 0
cot(z) = cos(z) / sin(z)
Zeros of cotangent: z = (2n+1)π/2, n ∈ ℤ
Theorems
Properties of Cotangent Function
Zeros of Trigonometric Functions
Suitable Grade Level
College Level
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