Which of the following statements shows how to calculate the reference angle for $\theta=\frac{11 \pi}{6}$ ?
The reference angle of an angle \(\theta\) is defined as the smallest positive acute angle formed between the terminal side of \(\theta\) and the x-axis. This angle must always be between \(0\) and \(\frac{\pi}{2}\). To find the reference angle, we must first identify the quadrant in which the angle \(\theta\) lies. The angle \(\theta = \frac{11\pi}{6}\) is expressed in radians. Given that a full rotation is \(2\pi = \frac{12\pi}{6}\), we can see that \(\frac{11\pi}{6}\) is slightly less than \(2\pi\). Specifically, \(\frac{3\pi}{2} = \frac{9\pi}{6}\). Thus, \(\frac{9\pi}{6} < \frac{11\pi}{6} < \frac{12\pi}{6}\), which means the terminal side of the angle lies in the fourth quadrant.
In trigonometry, the formulas for finding the reference angle \(\alpha\) depend on the quadrant in which \(\theta\) lies:
Since we have determined that \(\theta = \frac{11\pi}{6}\) lies in the fourth quadrant, we use the formula \(\alpha = 2\pi - \theta\).
Using the formula identified for the fourth quadrant:
\(\alpha = 2\pi - \frac{11\pi}{6}\)
To perform the subtraction, find a common denominator:
\(\alpha = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6}\)
The resulting angle \(\frac{\pi}{6}\) is an acute, positive angle. Thus, the correct expression to calculate the reference angle for \(\theta = \frac{11\pi}{6}\) is \(2\pi - \theta\).
Reference angles are fundamental in simplifying trigonometric calculations for any angle. For example, if an engineer is calculating the wave pattern of a signal at \(\frac{11\pi}{6}\) radians, they can use the reference angle \(\frac{\pi}{6}\) to find the sine or cosine value using familiar values from the first quadrant, such as \(\sin(\frac{11\pi}{6}) = -\sin(\frac{\pi}{6}) = -0.5\). This technique makes complex periodic functions much easier to manage.