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To solve the problem:

Given:
- \(\frac{\pi}{2} < \theta < \pi\), which means \(\theta\) is in the second quadrant.
- \(\sin(\theta) = \frac{1}{4}\).

We need to evaluate:
1. \(\cos(\theta)\)
2. \(\tan(\theta)\)

### Step-by-Step Solution

1. **Determine \(\cos(\theta)\):**

   We know that:
   \[
   \sin^2(\theta) + \cos^2(\theta) = 1
   \]
   Substitute \(\sin(\theta) = \frac{1}{4}\):
   \[
   \left(\frac{1}{4}\right)^2 + \cos^2(\theta) = 1
   \]
   \[
   \frac{1}{16} + \cos^2(\theta) = 1
   \]
   \[
   \cos^2(\theta) = 1 - \frac{1}{16} = \frac{15}{16}
   \]
   Taking the square root:
   \[
   \cos(\theta) = \pm \sqrt{\frac{15}{16}} = \pm \frac{\sqrt{15}}{4}
   \]
   
   Since \(\theta\) is in the second quadrant, where cosine is negative, we have:
   \[
   \cos(\theta) = -\frac{\sqrt{15}}{4}
   \]

2. **Determine \(\tan(\theta)\):**

   \[
   \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
   \]
   Substitute the values:
   \[
   \tan(\theta) = \frac{\frac{1}{4}}{-\frac{\sqrt{15}}{4}} = -\frac{1}{\sqrt{15}} = -\frac{\sqrt{15}}{15}
   \]

### Final Answers

1. \(\cos(\theta) = -\frac{\sqrt{15}}{4}\)
2. \(\tan(\theta) = -\frac{\sqrt{15}}{15}\)

Would you like further details on any step?

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Here are five related questions for practice:

1. What is \(\sin(2\theta)\) if \(\sin(\theta) = \frac{1}{4}\) and \(\theta\) is in the second quadrant?
2. If \(\theta\) is in the third quadrant, how would \(\cos(\theta)\) and \(\tan(\theta)\) change?
3. How would you find \(\sec(\theta)\) using the given information?
4. What is the value of \(\cot(\theta)\) given \(\sin(\theta) = \frac{1}{4}\) and \(\theta\) in the second quadrant?
5. Can you determine \(\cos^2(\theta) - \sin^2(\theta)\) from the given information?

**Tip:** When solving trigonometric identities, always pay attention to the quadrant, as it affects the sign of each trigonometric function.

Suppose \( \frac{\pi}{2} < \theta < \pi \), and \( \sin(\theta) = \frac{1}{4} \). Evaluate \( \cos(\theta) \) and \( \tan(\theta) \).

Learn how to evaluate \( \cos(\theta) \) and \( \tan(\theta) \) when \( \sin(\theta) = \frac{1}{4} \) and \( \theta \) is in the second quadrant. This problem covers key trigonometric identities, including the Pythagorean identity, with a step-by-step solution.

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