Math Problem Statement
Solution
Let's solve the problem step by step.
Part (1): Finding the Center of Symmetry of
Given:
First, simplify the function :
Thus,
A function has a center of symmetry at if .
For , we need to check if it has a center of symmetry.
Let be the center of symmetry. Then:
Since is an odd function:
For this equality to hold true, must be a point where is symmetrical, i.e., .
Thus, the center of symmetry is:
Therefore, the function has centers of symmetry at:
Part (2): Finding the Range of
Given:
We need to find the number of zeros of in the interval .
The zeros of occur when:
The sine function at:
We need 2024 zeros in the interval:
Let’s count the number of such zeros:
Given there are 2024 zeros:
Thus:
Therefore, the value of must be:
Summary
- The centers of symmetry of the function are , where .
- The value of such that the function has exactly 2024 zeros in the interval is:
Do you want further details or have any questions?
Here are some related questions you might want to consider:
- How do you determine the zeros of a trigonometric function?
- What are the general properties of sine and cosine functions?
- How do you solve trigonometric equations within a given interval?
- What is the significance of symmetry in trigonometric functions?
- How can you determine the period of a sine or cosine function?
Tip: When working with trigonometric functions, always remember their periodic nature and key values within one period. This can greatly simplify solving equations involving these functions.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Function Analysis
Formulas
f(x) = sin^2(x) + sin(x) + cos^2(x)
sin(x) = -1
Theorems
Properties of sine and cosine functions
Center of symmetry
Suitable Grade Level
Grades 10-12
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