Question

find the exact value of each of the remaining trigonometric functions of \\( \theta \\).
\\( \sec \theta = 6, \quad \tan \theta > 0 \\)
\\( \sin \theta = \frac { \sqrt { 3 5 } } { 6 } \\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
\\( \cos \theta = \frac { 1 } { 6 } \\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
\\( \tan \theta = \square \\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find $\cos\theta$

Since $\sec\theta = 6$, and $\sec\theta=\frac{1}{\cos\theta}$, then $\cos\theta=\frac{1}{6}$.

Step2: Find $\tan\theta$

We know the identity $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Given $\sin\theta = \frac{\sqrt{35}}{6}$ and $\cos\theta=\frac{1}{6}$.
Substitute the values into the formula: $\tan\theta=\frac{\frac{\sqrt{35}}{6}}{\frac{1}{6}}=\sqrt{35}$.

Answer:

$\cos\theta=\frac{1}{6}$, $\tan\theta=\sqrt{35}$