Using a trigonometric identity, it is found that the secant of the angle is given by:
[tex]\sec{\theta} = -\frac{8\sqrt{55}}{55}[/tex]
Which identity relates the sine and the cosine of an angle?
The sine and the cosine of an angle are related by the following identity:
[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]
In this problem, the sine is of -3/8, hence the cosine is found as follows:
[tex]\left(-\frac{3}{8}\right) + \cos^2{\theta} = 1[/tex]
[tex]\frac{9}{64} + \cos^2{\theta} = 1[/tex]
[tex]\cos^2{\theta} = \frac{55}{64}[/tex]
[tex]\cos{\theta} = \pm \sqrt{\frac{55}{64}}[/tex]
On the third quadrant, the cosine is negative, hence:
[tex]\cos{\theta} = -\frac{\sqrt{55}}{8}[/tex]
What is the secant of an angle?
The secant of an angle is given by one divided by the cosine, hence:
[tex]\sec{\theta} = \frac{1}{\cos{\theta}} = -\frac{1}{\frac{\sqrt{55}}{8}} = -\frac{8}{\sqrt{55}} \times \frac{\sqrt{55}}{\sqrt{55}} = -\frac{8\sqrt{55}}{55}[/tex]
More can be learned about trigonometric identities at https://brainly.com/question/26676095
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