Consider the following:
I.`tan((pi)/(6))`
II. `tan((3pi)/(4))`
III. `tan((5pi)/(4))`
IV. `tan((2pi)/(3))`
Which one of the following is the correct order?

Options

A. `I lt IV lt II lt III`
B. `IV lt II lt I lt III`
C. `IV lt II lt III lt I`
D. `I lt II lt III lt II`

Answer

To solve the problem, we need to calculate the values of the tangent function for each of the angles given and then arrange them in order from smallest to largest.

### Step-by-Step Solution:

1. **Calculate `tan(π/6)`**:
- We know that `tan(π/6)` is equivalent to `tan(30°)`.
- The value of `tan(30°)` is `1/√3`.
- Therefore, `tan(π/6) = 1/√3`.

2. **Calculate `tan(3π/4)`**:
- `3π/4` is equivalent to `135°`.
- The tangent of `135°` can be calculated as:
\[ \tan(135°) = \tan(90° + 45°) = -\cot(45°) = -1. \]
- Therefore, `tan(3π/4) = -1`.

3. **Calculate `tan(5π/4)`**:
- `5π/4` is equivalent to `225°`.
- The tangent of `225°` can be calculated as:
\[ \tan(225°) = \tan(180° + 45°) = \tan(45°) = 1. \]
- Therefore, `tan(5π/4) = 1`.

4. **Calculate `tan(2π/3)`**:
- `2π/3` is equivalent to `120°`.
- The tangent of `120°` can be calculated as:
\[ \tan(120°) = \tan(180° - 60°) = -\tan(60°) = -√3. \]
- Therefore, `tan(2π/3) = -√3`.

5. **Arrange the values**:
- Now we have the following values:
- `tan(π/6) = 1/√3 ≈ 0.577`
- `tan(3π/4) = -1`
- `tan(5π/4) = 1`
- `tan(2π/3) = -√3 ≈ -1.732`
- To order these from smallest to largest:
- `tan(2π/3) = -√3` (smallest)
- `tan(3π/4) = -1`
- `tan(π/6) = 1/√3`
- `tan(5π/4) = 1` (largest)

6. **Final Order**:
- The correct order from smallest to largest is:
- `tan(2π/3)`, `tan(3π/4)`, `tan(π/6)`, `tan(5π/4)`.
- This corresponds to the order: IV, II, I, III.

### Conclusion:
The correct order is IV, II, I, III.