- The sine function, \(\sin \theta\), corresponds to the y-coordinate of a point on the unit circle.
- \(\sin \theta\) is positive in quadrants I and II, zero on the x-axis, and negative in quadrants III and IV.
- Evaluating the given angles: \(\sin 0^{\circ} = 0\), \(\sin 120^{\circ} > 0\), \(\sin 270^{\circ} = -1\), and \(\sin 90^{\circ} > 0\).
- The only negative value occurs at \(\theta = 270^{\circ}\), so the answer is \(\boxed{270^{\circ}}\).
Explanation
- Analysis of the Sine Function and Quadrants
The sine function \(\sin \theta\) represents the y-coordinate of a point on the unit circle corresponding to the angle \(\theta\). The sign of \(\sin \theta\) is determined by the quadrant or axis in which the terminal side of the angle lies:
- \(\sin \theta > 0\) in Quadrant I (\(0^{\circ} < \theta < 90^{\circ}\)) and Quadrant II (\(90^{\circ} < \theta < 180^{\circ}\)).
- \(\sin \theta = 0\) at \(\theta = 0^{\circ}, 180^{\circ}, 360^{\circ}, \dots\)
- \(\sin \theta < 0\) in Quadrant III (\(180^{\circ} < \theta < 270^{\circ}\)) and Quadrant IV (\(270^{\circ} < \theta < 360^{\circ}\)).
- At \(\theta = 90^{\circ}\), \(\sin 90^{\circ} = 1\).
- At \(\theta = 270^{\circ}\), \(\sin 270^{\circ} = -1\).
- Evaluation of Given Angles
We evaluate \(\sin \theta\) for each of the given angles:
- For \(\theta = 0^{\circ}\): The point is \((1, 0)\), so \(\sin 0^{\circ} = 0\).
- For \(\theta = 120^{\circ}\): This angle is in the second quadrant, where sine is positive. \(\sin 120^{\circ} = \sin(180^{\circ} - 60^{\circ}) = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \approx 0.866 > 0\).
- For \(\theta = 270^{\circ}\): This angle corresponds to the point \((0, -1)\) on the unit circle. Therefore, \(\sin 270^{\circ} = -1\). Since \(-1 < 0\), the sine value is negative.
- For \(\theta = 90^{\circ}\): This angle corresponds to the point \((0, 1)\) on the unit circle. Therefore, \(\sin 90^{\circ} = 1 > 0\).
- Conclusion
Comparing our results, we find that:
- \(\sin 0^{\circ} = 0\) (not negative)
- \(\sin 120^{\circ} > 0\) (not negative)
- \(\sin 270^{\circ} = -1\) (negative)
- \(\sin 90^{\circ} > 0\) (not negative)
Thus, \(\theta = 270^{\circ}\) is the only angle among those provided where \(\sin \theta\) is negative. The correct value is \(\boxed{270^{\circ}}\).
Examples
Trigonometric functions are essential in modeling periodic phenomena, such as the height of tides over time or the vertical position of a person on a Ferris wheel. For instance, if an angle represents the rotation of a wheel, the sine function gives the vertical displacement relative to the center. Knowing where the sine value is negative helps engineers and analysts understand when an object is below the horizontal center line, which is critical for safety checks and design in mechanical systems.