To determine the cosecant of an angle given that the tangent of the angle is [tex]\(\frac{22}{9}\)[/tex], we can follow these steps:
1. Understanding the given information:
The tangent of an angle ([tex]\(\theta\)[/tex]) is given by:
[tex]\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{22}{9}
\][/tex]
2. Relationship between tangent, sine, and cosine:
We know the relationships:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
and the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
3. Express [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] using [tex]\(\tan \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}
\][/tex]
[tex]\[
\cos \theta = \frac{1}{\sqrt{1 + \tan^2 \theta}}
\][/tex]
4. Calculate [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
Given [tex]\(\tan \theta = \frac{22}{9}\)[/tex], we can compute:
[tex]\[
\tan^2 \theta = \left(\frac{22}{9}\right)^2 = \frac{484}{81}
\][/tex]
Thus,
[tex]\[
1 + \tan^2 \theta = 1 + \frac{484}{81} = \frac{81}{81} + \frac{484}{81} = \frac{565}{81}
\][/tex]
Then,
[tex]\[
\cos \theta = \frac{1}{\sqrt{1 + \tan^2 \theta}} = \frac{1}{\sqrt{\frac{565}{81}}} = \frac{1}{\frac{\sqrt{565}}{9}} = \frac{9}{\sqrt{565}}
\][/tex]
and
[tex]\[
\sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} = \frac{\frac{22}{9}}{\sqrt{\frac{565}{81}}} = \frac{\frac{22}{9}}{\frac{\sqrt{565}}{9}} = \frac{22}{\sqrt{565}}
\][/tex]
5. Determine the cosecant ([tex]\(\csc \theta\)[/tex]):
The cosecant is the reciprocal of the sine:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{22}{\sqrt{565}}} = \frac{\sqrt{565}}{22}
\][/tex]
Thus, the cosecant of the angle [tex]\(\theta\)[/tex] is:
[tex]\[
\boxed{\frac{\sqrt{565}}{22}}
\][/tex]
