We are given the following information to find $\theta$:

1. $\sin(\theta) = -0.3813$
2. $\tan(\theta) > 0$
3. $0 \leq \theta < 2\pi$

Step 1: Identify the possible quadrants

• Since $\sin(\theta)$ is negative, $\theta$ must be in either the third or fourth quadrant.
• Given $\tan(\theta) > 0$, the tangent is positive in the third quadrant (since both sine and cosine are negative in this quadrant).

Thus, $\theta$ must be in the third quadrant.

Step 2: Find the reference angle

We will first find the reference angle $\theta_{\text{ref}}$, which is the angle formed with the x-axis. The reference angle is the angle whose sine is the absolute value of $\sin(\theta)$.

$\theta_{\text{ref}} = \arcsin(0.3813)$

Let's calculate that.

$\theta_{\text{ref}} \approx \arcsin(0.3813) \approx 0.392 \, \text{radians}$

Step 3: Calculate $\theta$ in the third quadrant

In the third quadrant, the angle $\theta$ is given by:

$\theta = \pi + \theta_{\text{ref}}$

$\theta \approx \pi + 0.392 = 3.534 \, \text{radians}$

Thus, the solution is:

$\theta \approx 3.534 \, \text{radians}$

Final Answer:

$\theta \approx 3.534 \, \text{radians}$

Would you like further details, or do you have any questions?

Here are 5 related questions:

1. How do we determine the reference angle in general?
2. In which quadrants are the sine and cosine both negative?
3. How can we find the tangent given sine and cosine?
4. How does the unit circle help in determining angles?
5. What are the key properties of the trigonometric functions in each quadrant?

Tip: Always verify the quadrant and sign of trigonometric functions when solving for an angle.