If `tan theta=0.4`, when `theta` lies between `0^(@)` and `360^(@)`, write down the possible values of `theta` and `sin theta`.

Answer

To solve the problem where \( \tan \theta = 0.4 \) and \( \theta \) lies between \( 0^\circ \) and \( 360^\circ \), we will follow these steps:

### Step 1: Find the angles for \( \tan \theta = 0.4 \)

Since the tangent function is positive in the first and third quadrants, we will find the reference angle in the first quadrant and then determine the angles in both quadrants.

1. Calculate the reference angle using the arctangent function:
\[ \theta = \tan^{-1}(0.4) \]
Using a calculator, we find:
\[ \theta \approx 21.8^\circ \]

2. The angles in the first and third quadrants are:
- First Quadrant: \( \theta_1 = 21.8^\circ \)
- Third Quadrant: \( \theta_2 = 180^\circ + 21.8^\circ = 201.8^\circ \)

### Step 2: Write down the possible values of \( \theta \)

Thus, the possible values of \( \theta \) are:
\[ \theta \approx 21.8^\circ \quad \text{and} \quad \theta \approx 201.8^\circ \]

### Step 3: Find the values of \( \sin \theta \)

Now, we will find \( \sin \theta \) for both angles.

1. For \( \theta_1 = 21.8^\circ \):
\[ \sin(21.8^\circ) \approx 0.371 \]

2. For \( \theta_2 = 201.8^\circ \):
\[ \sin(201.8^\circ) = -\sin(21.8^\circ) \approx -0.371 \]

### Final Results

The possible values of \( \theta \) are:
\[ \theta \approx 21.8^\circ \quad \text{and} \quad \theta \approx 201.8^\circ \]

The corresponding values of \( \sin \theta \) are:
\[ \sin(21.8^\circ) \approx 0.371 \quad \text{and} \quad \sin(201.8^\circ) \approx -0.371 \]