Math Problem Statement

Find all solutions of cotx =sqrt(3)

Solution

We are tasked with solving the equation cot(x)=3\cot(x) = \sqrt{3}. To find all solutions, let's first express this in terms of the tangent function, as the cotangent is the reciprocal of the tangent:

cot(x)=1tan(x)\cot(x) = \frac{1}{\tan(x)}

Thus, the equation becomes:

1tan(x)=3\frac{1}{\tan(x)} = \sqrt{3}

Solving for tan(x)\tan(x), we multiply both sides by tan(x)\tan(x) and then by 3\sqrt{3}:

tan(x)=13\tan(x) = \frac{1}{\sqrt{3}}

Now, we need to find the general solution to tan(x)=13\tan(x) = \frac{1}{\sqrt{3}}. We know that:

tan(π6)=13\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}

So, one solution is x=π6x = \frac{\pi}{6}.

However, the tangent function has a period of π\pi. This means the general solution will be:

x=π6+nπwhere nZx = \frac{\pi}{6} + n\pi \quad \text{where} \ n \in \mathbb{Z}

Thus, the complete set of solutions is:

x=π6+nπ,nZx = \frac{\pi}{6} + n\pi, \quad n \in \mathbb{Z}

Summary of solutions:

x=π6+nπwhere nZx = \frac{\pi}{6} + n\pi \quad \text{where} \ n \in \mathbb{Z}

Let me know if you would like a more detailed explanation or have additional questions!


Here are some related questions for further study:

  1. What are the general solutions for cot(x)=3\cot(x) = -\sqrt{3}?
  2. How can we derive the general solution for a trigonometric function with period 2π2\pi, like sin(x)\sin(x)?
  3. What is the relationship between the unit circle and the values of the tangent and cotangent functions?
  4. How do we apply these periodic solutions in real-world applications, such as physics or engineering?
  5. 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 π\pi for tangent or 2π2\pi 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