Using a trigonometric identity, it is found that the tangent of the angle is of:
[tex]\tan{\theta} = \frac{2}{11}[/tex]
The sine and the cosine of the angle are related according to the following identity:
[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]
For this problem, the sine is given by:
[tex]\sin{\theta} = -\frac{2}{5\sqrt{5}}[/tex]
Hence the cosine is found as follows:
[tex]\left(-\frac{2}{5\sqrt{5}}\right)^2 + \cos^2{\theta} = 1[/tex]
[tex]\frac{4}{125} + \cos^2{\theta} = 1[/tex]
[tex]\cos^2{\theta} = \frac{121}{125}[/tex]
[tex]\cos{\theta} = \pm \sqrt{\frac{121}{125}}[/tex]
On the third quadrant the cosine is negative, hence:
[tex]\cos{\theta} = -\frac{11}{5\sqrt{5}}[/tex]
The tangent of an angle is given by the sine divided by the cosine, hence:
[tex]\tan{\theta} = \frac{-\frac{2}{5\sqrt{5}}}{-\frac{11}{5\sqrt{5}}} = \frac{2}{11}[/tex]
More can be learned about trigonometric identities at https://brainly.com/question/26676095
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