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]
