Question

An angle $\theta$ is such that $\sin \theta <0$ and $\cos \theta <0.$ In which quadrant does $\theta$ lie?

Short answer

Answer: Quadrant III

Step-by-step solution

Quadrant I

In the first quadrant, all coordinates are positive, which means both $\sin \theta$ and $\cos \theta$ are positive. So, $\theta$ cannot be in Quadrant I.

Quadrant II

In the second quadrant, the x-coordinate is negative while the y-coordinate is positive. Since sine represents the y-coordinate, we have $\sin \theta >0$. Since cosine represents the x-coordinate, we have $\cos \theta <0$. This means θ cannot be in Quadrant II.

Quadrant III

In the third quadrant, both the x and y coordinates are negative. This implies that both the sine and cosine functions are negative, meaning $\sin \theta <0$ and $\cos \theta <0$. Therefore, the angle $\theta$ must lie in Quadrant III.

Quadrant IV

In the fourth quadrant, the x-coordinate is positive while the y-coordinate is negative. Since sine represents the y-coordinate, we have $\sin \theta <0$. Since cosine represents the x-coordinate, we have $\cos \theta >0$. This means θ cannot be in Quadrant IV.

Conclusion

Based on the given conditions and the description of the sine and cosine functions in each quadrant, the angle $\theta$ lies in Quadrant III.

Key concepts

Quadrants of a Circle

In trigonometry, the circle is divided into four sections, known as quadrants. These quadrants help us determine the sine and cosine values for different angles. The circle is simply the coordinate system wrapping around 360 degrees, split into four 90-degree sections.
- **Quadrant I (0° to 90°):** Here, both coordinates are positive. This means sine (y-coordinate) and cosine (x-coordinate) are positive. - **Quadrant II (90° to 180°):** In this region, the x-coordinate is negative while the y-coordinate remains positive. Therefore, cosine is negative, but sine is positive. - **Quadrant III (180° to 270°):** Both x and y are negative, making both sine and cosine negative as well. - **Quadrant IV (270° to 360°):** The x-coordinate is positive while the y-coordinate is negative. Hence, cosine is positive, but sine is negative.
Understanding these sections is crucial when solving trigonometric problems based on an angle’s position on the unit circle.

Sine and Cosine Signs

Sine and cosine functions are pivotal trigonometric elements that reflect the coordinates of points on a unit circle. Each function's sign can differ across quadrants, greatly impacting angle computations.
**Sine Function: - Positive in Quadrants I and II. - Negative in Quadrants III and IV.**
**Cosine Function: - Positive in Quadrants I and IV. - Negative in Quadrants II and III.**
These signs serve as indicators to determine which quadrant an angle lies in, by analyzing whether sine and cosine are positive or negative. For instance, if both sine and cosine are negative, the angle resides in Quadrant III, as explained in our original solution.

Angle Properties

Angles, when viewed through the lens of trigonometry and a circle, exhibit specific properties depending on their quadrant location. Each angle's sine and cosine values reveal essential information about its position on the circle.
- **Angle in Quadrant I:** Here, angles produce positive values signifying both upward and rightward coordinates. - **Angle in Quadrant II:** Sine is positive, suggesting a positive y-value, but with a negative x-value reflecting leftward orientation. - **Angle in Quadrant III:** With negative sine and cosine values, angles show both downward and leftward coordinates. - **Angle in Quadrant IV:** Positive cosine indicates rightward positioning, while a negative sine value showcases downward movement.
These properties are derived from core trigonometric principles, assisting in understanding more complex angle-related problems. They act as a guide for determining the exact location of an angle based on the given sine and cosine values.