The expression given is \(\tan^{-1}(\tan(4\pi/3))\), which means we need to find the inverse tangent (or arctangent) of \(\tan(4\pi/3)\).

### Step 1: Simplify \(\tan(4\pi/3)\)

We know that:
- \(4\pi/3\) is in the third quadrant, and the tangent function is positive in this quadrant.
- The reference angle for \(4\pi/3\) is \(\pi/3\).

So:
\[
\tan(4\pi/3) = \tan(\pi/3) = \sqrt{3}
\]
However, because \(4\pi/3\) is in the third quadrant, the tangent value will be negative:
\[
\tan(4\pi/3) = -\sqrt{3}
\]

### Step 2: Find the inverse tangent

The inverse tangent function, \(\tan^{-1}(x)\), gives an angle in the principal range of \((- \frac{\pi}{2}, \frac{\pi}{2})\). We need to find the angle whose tangent is \(-\sqrt{3}\).

The angle that satisfies this is:
\[
\tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3}
\]

Thus:
\[
\tan^{-1}(\tan(4\pi/3)) = -\frac{\pi}{3}
\]

### Final Answer:
\[
\tan^{-1}(\tan(4\pi/3)) = -\frac{\pi}{3}
\]

Would you like further details on any of these steps?

### Related Questions:
1. What is the general solution for \(\tan(\theta) = x\)?
2. How does the arctangent function behave outside its principal range?
3. How do you find reference angles for angles in different quadrants?
4. What are the properties of the tangent function in all four quadrants?
5. Can you explain the domain and range of the \(\tan^{-1}\) function?

#### Tip:
Always remember that the principal value of the inverse trigonometric functions is restricted to specific ranges, which helps determine unique solutions.

Learn how to solve tan^−1(tan(4π/3)) using inverse trigonometric functions, quadrant-based angle simplification, and the principal value theorem. This detailed step-by-step solution is suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation