We are asked to determine the trigonometric functions for:
[tex]\theta=\frac{23\pi}{6}[/tex]
To determine the trigonometric functions we need to determine the equivalent angle that is between 0 and 2pi. To do that we will subtract 2pi from the given angle:
[tex]\frac{23\pi}{6}-2\pi=\frac{11\pi}{6}[/tex]
The equivalent angle is 11pi/6. In the unit circle this angle is:
The end-point of this angle is:
[tex](\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]
Since the x-component of the end-point is the cosine, we have:
[tex]cos(\frac{23\pi}{6})=\frac{\sqrt{3}}{2}[/tex]
The y-coordinate is the sine, therefore:
[tex]sin(\frac{23\pi}{6})=-\frac{1}{2}[/tex]
To determine the tangent we use the following relationship:
[tex]tanx=\frac{sinx}{cosx}[/tex]
Substituting we get:
[tex]tan(\frac{23\pi}{6})=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}[/tex]
Simplifying we get:
[tex]\begin{gathered} tan(\frac{23\pi}{6})=\frac{-1}{\sqrt{3}} \\ \end{gathered}[/tex]
To determine the secant we use the following relationship:
[tex]secx=\frac{1}{cosx}[/tex]
Substituting we get:
[tex]sec(\frac{23\pi}{6})=\frac{1}{\frac{\sqrt{3}}{2}}[/tex]
Simplifying we get:
[tex]sec(\frac{23\pi}{6})=\frac{2}{\sqrt{3}}[/tex]
To determine the cosecant we use the following relationship:
[tex]cscx=\frac{1}{sinx}[/tex]
Substituting we get:
[tex]csc(\frac{23\pi}{6})=\frac{1}{-\frac{1}{2}}[/tex]
Simplifying we get:
[tex]csc(\frac{23\pi}{6})=-2[/tex]
Finally, for the cotangent we use:
[tex]ctgx=\frac{1}{tanx}[/tex]
Substituting we get:
[tex]ctgx=\frac{1}{\frac{-1}{\sqrt{3}}}[/tex]
Simplifying:
[tex]ctg(\frac{23\pi}{6})=-\sqrt{3}[/tex]
