Using the Pythagorean identity, the value of the cosine ratio is [tex]\cos(\theta_1) = \frac{84}{85}[/tex]
How to determine the cosine ratio?
The given parameter is:
[tex]\sin(\theta_1) = -\frac{13}{85}[/tex]
By the Pythagorean identity, we have:
[tex]\sin^2(\theta_1) + \cos^2(\theta_1) = 1[/tex]
So, we have:
[tex](-\frac{13}{85})^2 + \cos^2(\theta_1) = 1[/tex]
This gives
[tex]\cos^2(\theta_1) = 1 - (-\frac{13}{85})^2[/tex]
Evaluate
[tex]\cos^2(\theta_1) = 1 - \frac{169}{7225}[/tex]
Take LCM
[tex]\cos^2(\theta_1) = \frac{7225 -169}{7225}[/tex]
This gives
[tex]\cos^2(\theta_1) = \frac{7056}{7225}[/tex]
Take the square root of both sides
[tex]\cos(\theta_1) = \pm \frac{84}{85}[/tex]
Cosine is positive in the fourth quadrant.
So, we have:
[tex]\cos(\theta_1) = \frac{84}{85}[/tex]
Hence, the cosine value is [tex]\cos(\theta_1) = \frac{84}{85}[/tex]
Read more about Pythagorean identity at:
https://brainly.com/question/1969941
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