Respuesta :
Using the relation between the sine and the cosine of an angle, it is found that:
[tex]\cos{\theta} = -\frac{2\sqrt{2}}{3}[/tex]
What is the relation between the sine and the cosine of an angle?
The sine squared of an angle, plus the cosine squared of the same angle, is equals to one, hence:
[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]
In this problem, we have that the sine is given by:
[tex]\sin{\theta} = -\frac{1}{3}[/tex]
Hence:
[tex]\left(-\frac{1}{3}\right) + \cos^2{\theta} = 1[/tex]
[tex]\cos^2{\theta} = 1 - \frac{1}{9}[/tex]
[tex]\cos^2{\theta} = \frac{8}{9}[/tex]
[tex]\cos{\theta} = \pm \sqrt{\frac{8}{9}}[/tex]
[tex]\cos{\theta} = \pm \frac{2\sqrt{2}}{3}[/tex]
We have that pi < theta < 1.5pi, hence the angle is in the third quadrant, where the cosine is negative, thus:
[tex]\cos{\theta} = -\frac{2\sqrt{2}}{3}[/tex]
More can be learned about the relation between the sine and the cosine of an angle at https://brainly.com/question/27692818
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