Answer :
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].
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].
