We are asked to find $\sin(\theta)$, given that:

$\tan(\theta) = 3 \quad \text{and} \quad \pi < \theta < \frac{3\pi}{2}$

Step 1: Identify the quadrant.

The condition $\pi < \theta < \frac{3\pi}{2}$ means that $\theta$ is in the third quadrant. In this quadrant:

• $\sin(\theta)$ is negative.
• $\cos(\theta)$ is negative.
• $\tan(\theta)$ is positive.

Step 2: Use the relationship between $\tan(\theta)$ and the sides of a right triangle.

Since $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = 3$, we can interpret this as a ratio of the opposite side to the adjacent side of a right triangle:

• Opposite side: 3
• Adjacent side: 1

Now, let's use the Pythagorean theorem to find the hypotenuse: $r = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}$

Step 3: Find $\sin(\theta)$.

In a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Thus, $\sin(\theta) = \frac{3}{\sqrt{10}}$ Since $\theta$ is in the third quadrant, $\sin(\theta)$ must be negative: $\sin(\theta) = -\frac{3}{\sqrt{10}} = -\frac{3\sqrt{10}}{10}$

Thus, the value of $\sin(\theta)$ is $-\frac{3\sqrt{10}}{10}$.

Would you like more details on any step?

Here are 5 related questions to expand on the concept:

1. How can you find $\cos(\theta)$ in this scenario?
2. What is the relationship between the Pythagorean identity and trigonometric functions?
3. How do the signs of trigonometric functions change in each quadrant?
4. Can you verify this result using the unit circle?
5. How do you express $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$?

Tip: Understanding the signs of trigonometric functions in different quadrants helps simplify calculations for all trigonometric identities.