The image shows a trigonometry problem where you're given that $\sin(\theta) = \frac{5}{6}$ and $\theta$ is in the first quadrant. The goal is to find $\cos(\theta)$, and you have attempted $\cos(\theta) = \frac{\sqrt{34}}{6}$, which was marked incorrect. The instructions suggest expressing the answer as a reduced radical.

Step-by-step Solution:

1. Use the Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$ Since $\sin(\theta) = \frac{5}{6}$, we can square this value: $\left(\frac{5}{6}\right)^2 = \frac{25}{36}$ So, the equation becomes: $\cos^2(\theta) = 1 - \frac{25}{36} = \frac{36}{36} - \frac{25}{36} = \frac{11}{36}$

2. Solve for $\cos(\theta)$: Taking the square root of both sides: $\cos(\theta) = \sqrt{\frac{11}{36}} = \frac{\sqrt{11}}{6}$

Since $\theta$ is in the first quadrant, $\cos(\theta)$ is positive, so the correct answer is: $\cos(\theta) = \frac{\sqrt{11}}{6}$

Final Answer:

$\cos(\theta) = \frac{\sqrt{11}}{6}$

This should be the correct simplified radical form for the cosine of $\theta$.

Do you have any questions or would you like further clarification on this?

Here are 5 related questions to explore further:

1. How would you find $\tan(\theta)$ given $\sin(\theta) = \frac{5}{6}$?
2. What would change if $\theta$ were in the second quadrant?
3. How do you simplify radicals like $\sqrt{34}$?
4. How can the Pythagorean identity be applied to secant and cosecant functions?
5. What is the relationship between sine and cosine in terms of the unit circle?

Tip: When simplifying radicals, always check if the square root can be broken down further by factoring out perfect squares.