θ is an acute angle. Find the value of θ in radians.
cot(θ)=
3
Write your answer in simplified, rationalized form. Do not round.

Given that \(\theta\) is an acute angle, and \(\cot(\theta) = 3\), we need to find the value of \(\theta\) in radians.

### Step 1: Relationship between cotangent and tangent
We know that \(\cot(\theta)\) is the reciprocal of \(\tan(\theta)\):
\[
\cot(\theta) = \frac{1}{\tan(\theta)}
\]
Given \(\cot(\theta) = 3\), we can write:
\[
\tan(\theta) = \frac{1}{3}
\]

### Step 2: Use inverse tangent
To find \(\theta\), we need to use the inverse tangent function:
\[
\theta = \tan^{-1}\left(\frac{1}{3}\right)
\]

This is the exact expression for \(\theta\), but in order to give the value in radians, we write the answer in terms of the inverse tangent function:
\[
\theta = \tan^{-1}\left(\frac{1}{3}\right)
\]

This is the simplified, rationalized form of the solution.

### Final Answer:
\[
\theta = \tan^{-1}\left(\frac{1}{3}\right) \text{ radians}
\]

Would you like a deeper explanation of any step?

Here are some related questions:
1. What is the relationship between \(\cot(\theta)\) and \(\sin(\theta)\)?
2. How do you convert from degrees to radians?
3. How can the Pythagorean identity be used to relate \(\sin(\theta)\) and \(\cos(\theta)\)?
4. What is the value of \(\theta\) in degrees when \(\cot(\theta) = 3\)?
5. How does the value of \(\theta\) change if \(\cot(\theta)\) is negative?

**Tip:** Always remember that \(\tan^{-1}(x)\) gives the angle whose tangent is \(x\), and the output of \(\tan^{-1}(x)\) is typically in radians for most applications.

θ is an acute angle. Find the value of θ in radians. cot(θ) = 3. Write your answer in simplified, rationalized form. Do not round.

Solve for θ in radians when cot(θ) = 3. This trigonometry problem covers the relationship between cotangent and tangent, and how to use the inverse tangent function. Suitable for students in Grades 9-11, with step-by-step solutions to find the value of θ in simplified, rationalized form.

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