Questions: Find the exact value of the trigonometric function at the given real number. (a) csc (11π/6) (b) sec (-5π/6) (c) cot (-5π/6)

Solution Steps

To find the exact values of the given trigonometric functions, we can use the unit circle and the definitions of the trigonometric functions.

1. For $\csc \left(\frac{11 \pi}{6}\right)$, we need to find the sine of $\frac{11 \pi}{6}$ and then take its reciprocal.
2. For $\sec \left(-\frac{5 \pi}{6}\right)$, we need to find the cosine of $-\frac{5 \pi}{6}$ and then take its reciprocal.
3. For $\cot \left(-\frac{5 \pi}{6}\right)$, we need to find the tangent of $-\frac{5 \pi}{6}$ and then take its reciprocal.

To find \(\csc \left(\frac{11 \pi}{6}\right)\), we first determine \(\sin \left(\frac{11 \pi}{6}\right)\). The angle \(\frac{11 \pi}{6}\) is in the fourth quadrant, where sine is negative. Specifically, \(\sin \left(\frac{11 \pi}{6}\right) = -\frac{1}{2}\).

\[ \csc \left(\frac{11 \pi}{6}\right) = \frac{1}{\sin \left(\frac{11 \pi}{6}\right)} = \frac{1}{-\frac{1}{2}} = -2 \]

To find \(\sec \left(-\frac{5 \pi}{6}\right)\), we first determine \(\cos \left(-\frac{5 \pi}{6}\right)\). The angle \(-\frac{5 \pi}{6}\) is in the third quadrant, where cosine is negative. Specifically, \(\cos \left(-\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2}\).

\[ \sec \left(-\frac{5 \pi}{6}\right) = \frac{1}{\cos \left(-\frac{5 \pi}{6}\right)} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \]

To find \(\cot \left(-\frac{5 \pi}{6}\right)\), we first determine \(\tan \left(-\frac{5 \pi}{6}\right)\). The angle \(-\frac{5 \pi}{6}\) is in the third quadrant, where tangent is positive. Specifically, \(\tan \left(-\frac{5 \pi}{6}\right) = \frac{1}{\sqrt{3}}\).

\[ \cot \left(-\frac{5 \pi}{6}\right) = \frac{1}{\tan \left(-\frac{5 \pi}{6}\right)} = \sqrt{3} \]

Final Answer