Math Problem Statement
Find all solutions of cotx =sqrt(3)
Solution
We are tasked with solving the equation . To find all solutions, let's first express this in terms of the tangent function, as the cotangent is the reciprocal of the tangent:
Thus, the equation becomes:
Solving for , we multiply both sides by and then by :
Now, we need to find the general solution to . We know that:
So, one solution is .
However, the tangent function has a period of . This means the general solution will be:
Thus, the complete set of solutions is:
Summary of solutions:
Let me know if you would like a more detailed explanation or have additional questions!
Here are some related questions for further study:
- What are the general solutions for ?
- How can we derive the general solution for a trigonometric function with period , like ?
- What is the relationship between the unit circle and the values of the tangent and cotangent functions?
- How do we apply these periodic solutions in real-world applications, such as physics or engineering?
- What are some common errors when solving trigonometric equations involving periodic functions?
Tip: When solving trigonometric equations, always consider the periodic nature of the functions (such as for tangent or for sine and cosine) to ensure you find all possible solutions.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Periodic Functions
Cotangent and Tangent Relationship
Formulas
cot(x) = 1/tan(x)
tan(x) = 1/sqrt(3)
General solution for tangent: x = arctan(value) + nπ, where n ∈ Z
Theorems
Periodic Nature of Tangent Function
Suitable Grade Level
Grades 10-12