- Use the odd property of the sine function: \(\sin(-135^{\circ}) = -\sin(135^{\circ})\).
- Determine the reference angle by subtracting the angle from the x-axis: \(180^{\circ} - 135^{\circ} = 45^{\circ}\).
- Calculate the sine value using the reference angle: \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\).
- Combine the results: \(\sin(-135^{\circ}) = -\frac{\sqrt{2}}{2}\).
- Final Answer: Reference angle is \(\boxed{45}^{\circ}\); \(\sin(-135)^{\circ} = \boxed{-\frac{\sqrt{2}}{2}}\).
Explanation
- Applying Properties of the Sine Function
The goal is to determine the exact value of \(\sin(-135^{\circ})\) and identify its reference angle. The sine function is an odd function, meaning \(\sin(-\theta) = -\sin(\theta)\) for any angle \(\theta\). This property allows us to simplify the expression: \(\sin(-135^{\circ}) = -\sin(135^{\circ})\).
- Identifying the Reference Angle
To find \(\sin(135^{\circ})\), we locate the angle \(135^{\circ}\) on the unit circle. Since \(90^{\circ} < 135^{\circ} < 180^{\circ}\), the angle lies in the second quadrant. The reference angle \(\theta\) is defined as the smallest acute angle the terminal side makes with the x-axis. For an angle in the second quadrant, the reference angle is calculated as \(180^{\circ} - \alpha\), where \(\alpha = 135^{\circ}\). Therefore, the reference angle is \(\theta = 180^{\circ} - 135^{\circ} = 45^{\circ}\).
- Evaluating the Sine of the Reference Angle
In the second quadrant, the sine function is positive. Using the reference angle identified in the previous step, we have \(\sin(135^{\circ}) = \sin(45^{\circ})\). We know from the unit circle properties that \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\). Thus, \(\sin(135^{\circ}) = \frac{\sqrt{2}}{2}\).
- Final Calculation and Conclusion
Finally, we substitute the value back into our initial expression: \(\sin(-135^{\circ}) = -\sin(135^{\circ}) = -\left(\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{2}\). The reference angle is \(45^{\circ}\). Therefore, the reference angle \(\theta\) is \(\boxed{45}^{\circ}\) and the value of \(\sin(-135)^{\circ}\) is \(\boxed{-\frac{\sqrt{2}}{2}}\).
Examples
Understanding trigonometric functions and their reference angles is fundamental in physics and engineering, particularly when analyzing periodic phenomena like sound waves or alternating currents. For instance, if an electrical engineer needs to determine the instantaneous voltage of an AC signal at a specific phase angle, they use trigonometric identities to resolve the signal into its components. Knowing how to relate an angle in any quadrant back to a reference angle in the first quadrant allows for simple and accurate calculations of these values.