When given the equation \(\tan \theta = \frac{a}{b}\), the goal is to find the value of \(\theta\).

Understanding the Tangent Function
- The tangent function, \(\tan \theta\), represents the ratio of the opposite side to the adjacent side in a right triangle.
- It can also be expressed as:
- \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

Finding \(\theta\)
- To isolate \(\theta\), you will use the arctangent function (also known as the inverse tangent function).
- The formula to find \(\theta\) is:
- \(\theta = \tan^{-1} \left(\frac{a}{b}\right)\)

Steps to Calculate \(\theta\)
- Identify the values of \(a\) and \(b\).
- Substitute these values into the equation:
- \(\theta = \tan^{-1} \left(\frac{a}{b}\right)\)
- Use a calculator or mathematical tables to determine the value of \(\theta\).

Considerations
- Ensure \(b\) is not equal to zero, as division by zero is undefined.
- The value of \(\theta\) will typically be in radians or degrees, depending on your calculator settings.

Conclusion
- The solution to \(\tan \theta = \frac{a}{b}\) can be effectively determined using the inverse tangent function.
- This method provides a direct approach to solving for angles in trigonometric applications, relevant in various fields, including physics, engineering, and navigation.