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Given [tex]\sin \theta = \frac{6}{11}[/tex] and [tex]\sec \theta \ \textless \ 0[/tex], find [tex]\cos \theta[/tex] and [tex]\tan \theta[/tex].

a. [tex]\cos \theta = \frac{-\sqrt{85}}{11}, \tan \theta = \frac{-6 \sqrt{85}}{85}[/tex]

b. [tex]\cos \theta = \frac{\sqrt{85}}{11}, \tan \theta = \frac{6 \sqrt{85}}{85}[/tex]

c. [tex]\cos \theta = \frac{85}{11}, \tan \theta = \frac{11}{85}[/tex]

d. [tex]\cos \theta = \frac{-\sqrt{85}}{11}, \tan \theta = \frac{-\sqrt{85}}{6}[/tex]

Please select the best answer from the choices provided:
A, B, C, or D



Answer :

GinnyAnswer
To find [tex]\(\cos \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] given [tex]\(\sin \theta = \frac{6}{11}\)[/tex] and [tex]\(\sec \theta < 0\)[/tex], let's proceed with the steps.

1. Find [tex]\(\cos \theta\)[/tex] using the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute the given value of [tex]\(\sin \theta\)[/tex]:
[tex]\[ \left( \frac{6}{11} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{36}{121} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{36}{121} \][/tex]
[tex]\[ \cos^2 \theta = \frac{121}{121} - \frac{36}{121} \][/tex]
[tex]\[ \cos^2 \theta = \frac{85}{121} \][/tex]
[tex]\[ \cos \theta = \pm \frac{\sqrt{85}}{11} \][/tex]

2. Determine the sign of [tex]\(\cos \theta\)[/tex]:
Since [tex]\(\sec \theta < 0\)[/tex], it implies that [tex]\(\cos \theta\)[/tex] must be negative because [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex]. Thus,
[tex]\[ \cos \theta = -\frac{\sqrt{85}}{11} \][/tex]

3. Find [tex]\(\tan \theta\)[/tex] using the relationship:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Substitute the known values:
[tex]\[ \tan \theta = \frac{\frac{6}{11}}{-\frac{\sqrt{85}}{11}} \][/tex]
[tex]\[ \tan \theta = \frac{6}{-\sqrt{85}} \][/tex]
[tex]\[ \tan \theta = -\frac{6}{\sqrt{85}} \][/tex]
Rationalize the denominator:
[tex]\[ \tan \theta = -\frac{6 \sqrt{85}}{85} \][/tex]

So, the correct values are:
[tex]\[ \cos \theta = -\frac{\sqrt{85}}{11}, \quad \tan \theta = -\frac{6 \sqrt{85}}{85} \][/tex]

Thus, the correct answer is:
a. [tex]\(\cos \theta = \frac{-\sqrt{85}}{11}, \tan \theta = \frac{-6 \sqrt{85}}{85}\)[/tex]

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