To find the cosecant of an angle given the tangent, follow these steps:
1. Understand the given information:
- Tangent ([tex]\(\tan(\theta)\)[/tex]) of the angle is [tex]\(\frac{22}{9}\)[/tex].
- In a right triangle, [tex]\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex].
2. Identify the sides of the triangle:
- Let the opposite side be 22 and the adjacent side be 9 (as indicated by the tangent value).
3. Calculate the hypotenuse:
- Use the Pythagorean theorem which states [tex]\(\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2\)[/tex].
[tex]\[
\text{hypotenuse} = \sqrt{22^2 + 9^2}
\][/tex]
[tex]\[
\text{hypotenuse} = \sqrt{484 + 81}
\][/tex]
[tex]\[
\text{hypotenuse} = \sqrt{565}
\][/tex]
[tex]\[
\text{hypotenuse} \approx 23.769728648009426
\][/tex]
4. Determine the sine of the angle:
- Sine ([tex]\(\sin(\theta)\)[/tex]) is defined as [tex]\(\frac{\text{opposite}}{\text{hypotenuse}}\)[/tex].
[tex]\[
\sin(\theta) = \frac{22}{23.769728648009426}
\][/tex]
[tex]\[
\sin(\theta) \approx 0.9255469562056767
\][/tex]
5. Calculate the cosecant of the angle:
- Cosecant ([tex]\(\csc(\theta)\)[/tex]) is the reciprocal of sine.
[tex]\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\][/tex]
[tex]\[
\csc(\theta) = \frac{1}{0.9255469562056767}
\][/tex]
[tex]\[
\csc(\theta) \approx 1.0804422112731558
\][/tex]
Thus, the cosecant ([tex]\(\csc(\theta)\)[/tex]) of the angle is approximately [tex]\(1.0804422112731558\)[/tex].
