Answer:
Second answer
Step-by-step explanation:
We are given [tex]\displaystyle \large{\sec \theta = -3}[/tex] and [tex]\displaystyle \large{\sin \theta > 0}[/tex]. What we have to find are [tex]\displaystyle \large{\cot \theta}[/tex] and [tex]\displaystyle \large{\cos \theta}[/tex].
First, convert [tex]\displaystyle \large{\sec \theta}[/tex] to [tex]\displaystyle \large{\frac{1}{\cos \theta}}[/tex] via trigonometric identity. That gives us a new equation in form of [tex]\displaystyle \large{\cos \theta}[/tex]:
[tex]\displaystyle \large{\frac{1}{\cos \theta} = -3}[/tex]
Multiply [tex]\displaystyle \large{\cos \theta}[/tex] both sides to get rid of the denominator.
[tex]\displaystyle \large{\frac{1}{\cos \theta} \cdot \cos \theta = -3 \cos \theta}\\\displaystyle \large{1=-3 \cos \theta}[/tex]
Then divide both sides by -3 to get [tex]\displaystyle \large{\cos \theta}[/tex].
Hence, [tex]\displaystyle \large{\boxed{\cos \theta = - \frac{1}{3}}}[/tex]
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Next, to find [tex]\displaystyle \large{\cot \theta}[/tex], convert it to [tex]\displaystyle \large{\frac{1}{\tan \theta}}[/tex] via trigonometric identity. Then we have to convert [tex]\displaystyle \large{\tan \theta}[/tex] to [tex]\displaystyle \large{\frac{\sin \theta}{\cos \theta}}[/tex] via another trigonometric identity. That gives us:
[tex]\displaystyle \large{\frac{1}{\frac{\sin \theta}{\cos \theta}}}\\\displaystyle \large{\frac{\cos \theta}{\sin \theta}[/tex]
It seems that we do not know what [tex]\displaystyle \large{\sin \theta}[/tex] is but we can find it by using the identity [tex]\displaystyle \large{\sin \theta = \sqrt{1-\cos ^2 \theta}}[/tex] for [tex]\displaystyle \large{\sin \theta > 0}[/tex].
From [tex]\displaystyle \large{\cos \theta = -\frac{1}{3}}[/tex] then [tex]\displaystyle \large{\cos ^2 \theta = \frac{1}{9}}[/tex].
Therefore:
[tex]\displaystyle \large{\sin \theta=\sqrt{1-\frac{1}{9}}}\\\displaystyle \large{\sin \theta = \sqrt{\frac{9}{9}-\frac{1}{9}}}\\\displaystyle \large{\sin \theta = \sqrt{\frac{8}{9}}}[/tex]
Then use the surd property to evaluate the square root.
Hence, [tex]\displaystyle \large{\boxed{\sin \theta=\frac{2\sqrt{2}}{3}}}[/tex]
Now that we know what [tex]\displaystyle \large{\sin \theta}[/tex] is. We can evaluate [tex]\displaystyle \large{\frac{\cos \theta}{\sin \theta}}[/tex] which is another form or identity of [tex]\displaystyle \large{\cot \theta}[/tex].
From the boxed values of [tex]\displaystyle \large{\cos \theta}[/tex] and [tex]\displaystyle \large{\sin \theta}[/tex]:-
[tex]\displaystyle \large{\cot \theta = \frac{\cos \theta}{\sin \theta}}\\\displaystyle \large{\cot \theta = \frac{-\frac{1}{3}}{\frac{2\sqrt{2}}{3}}}\\\displaystyle \large{\cot \theta=-\frac{1}{3} \cdot \frac{3}{2\sqrt{2}}}\\\displaystyle \large{\cot \theta=-\frac{1}{2\sqrt{2}}[/tex]
Then rationalize the value by multiplying both numerator and denominator with the denominator.
[tex]\displaystyle \large{\cot \theta = -\frac{1 \cdot 2\sqrt{2}}{2\sqrt{2} \cdot 2\sqrt{2}}}\\\displaystyle \large{\cot \theta = -\frac{2\sqrt{2}}{8}}\\\displaystyle \large{\cot \theta = -\frac{\sqrt{2}}{4}}[/tex]
Hence, [tex]\displaystyle \large{\boxed{\cot \theta = -\frac{\sqrt{2}}{4}}}[/tex]
Therefore, the second choice is the answer.
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Summary
[tex]\displaystyle \large{\sec \theta = \frac{1}{\cos \theta}}\\ \displaystyle \large{\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}}\\ \displaystyle \large{\sin \theta = \sqrt{1-\cos ^2 \theta} \ \ \ (\sin \theta > 0)}\\ \displaystyle \large{\tan \theta = \frac{\sin \theta}{\cos \theta}}[/tex]
[tex]\displaystyle \large{\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}}[/tex]
Let me know in the comment if you have any questions regarding this question or for clarification! Hope this helps as well.