The conditions given are:

• tan(θ) < 0
• sin(θ) < 0

We will analyze these conditions with respect to the four quadrants of the unit circle.

1. tan(θ) < 0:

The tangent of an angle is the ratio of the sine to the cosine: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ For $\tan(\theta) < 0$, this means that the sine and cosine must have opposite signs. Specifically:

• In the 2nd quadrant, sine is positive and cosine is negative, so $\tan(\theta) < 0$.
• In the 4th quadrant, sine is negative and cosine is positive, so $\tan(\theta) < 0$.

Thus, the condition $\tan(\theta) < 0$ restricts $\theta$ to the 2nd and 4th quadrants.

2. sin(θ) < 0:

The sine of an angle is negative in the 3rd and 4th quadrants, where the y-coordinate of the point on the unit circle is negative.

Combining the Conditions:

• $\tan(\theta) < 0$ suggests $\theta$ is in the 2nd or 4th quadrants.
• $\sin(\theta) < 0$ restricts $\theta$ to the 3rd or 4th quadrants.

The only quadrant that satisfies both conditions (where $\tan(\theta) < 0$ and $\sin(\theta) < 0$) is the 4th quadrant.

Conclusion:

The angle $\theta$ must be in the 4th quadrant to satisfy both $\tan(\theta) < 0$ and $\sin(\theta) < 0$.

---

Would you like a more detailed explanation or have further questions?

Here are some related questions to expand on this topic:

1. What are the signs of sine, cosine, and tangent in each quadrant?
2. How do you graph the sine and tangent functions?
3. How can you determine the exact value of $\theta$ given specific conditions on sine or tangent?
4. What is the reference angle and how does it relate to these conditions?
5. How do the unit circle and trigonometric identities help in solving such problems?

Tip: When determining the quadrant of an angle based on trigonometric signs, always visualize the unit circle and recall that sine is related to the vertical axis, cosine to the horizontal axis, and tangent to the ratio of these two.