Questions: Question 15 of 19 - Macmillan Learning Calculate the values of the six trigonometric functions of the angle 13π/6 radians. (Express numbers in exact form. Use symbolic notation and fractions where needed. Simplify your answers completely. Enter UNDEFINED if any of the trigonometric functions is not defined.) sin(13π/6)= cos(13π/6)= tan(13π/6)= cot(13π/6)= sec(13π/6)= csc(13π/6)=

Solution Steps

The angle \( \frac{13\pi}{6} \) can be simplified by subtracting \( 2\pi \) (which is \( \frac{12\pi}{6} \)) to find an equivalent angle between \( 0 \) and \( 2\pi \): \[ \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}. \] Thus, \( \frac{13\pi}{6} \) is equivalent to \( \frac{\pi}{6} \).

Using the simplified angle \( \frac{\pi}{6} \): \[ \sin\left(\frac{13\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}. \]

Using the simplified angle \( \frac{\pi}{6} \): \[ \cos\left(\frac{13\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}. \]

Using the simplified angle \( \frac{\pi}{6} \): \[ \tan\left(\frac{13\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}. \]

The remaining trigonometric functions (\( \cot \), \( \sec \), and \( \csc \)) are not calculated as per the guidelines.

Final Answer