Evaluate the following expression:
`sin((pi)/6+"cos"^(-1)1/4)`

Explanation

To evaluate the expression \( \sin\left(\frac{\pi}{6} + \cos^{-1}\left(\frac{1}{4}\right)\right) \), we can follow these steps:

### Step 1: Identify the components of the expression
We have:
- \( a = \frac{\pi}{6} \)
- \( b = \cos^{-1}\left(\frac{1}{4}\right) \)

### Step 2: Use the sine addition formula
The sine addition formula states:
\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]
Applying this to our expression:
\[ \sin\left(\frac{\pi}{6} + \cos^{-1}\left(\frac{1}{4}\right)\right) = \sin\left(\frac{\pi}{6}\right)\cos\left(\cos^{-1}\left(\frac{1}{4}\right)\right) + \cos\left(\frac{\pi}{6}\right)\sin\left(\cos^{-1}\left(\frac{1}{4}\right)\right) \]

### Step 3: Calculate \( \sin\left(\frac{\pi}{6}\right) \) and \( \cos\left(\frac{\pi}{6}\right) \)
We know:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

### Step 4: Calculate \( \cos\left(\cos^{-1}\left(\frac{1}{4}\right)\right) \) and \( \sin\left(\cos^{-1}\left(\frac{1}{4}\right)\right) \)
Using the property of inverse trigonometric functions:
\[ \cos\left(\cos^{-1}\left(\frac{1}{4}\right)\right) = \frac{1}{4} \]
To find \( \sin\left(\cos^{-1}\left(\frac{1}{4}\right)\right) \), we can use the Pythagorean identity:
\[ \sin^2(b) + \cos^2(b) = 1 \]
Let \( \sin(b) = y \). Then:
\[ y^2 + \left(\frac{1}{4}\right)^2 = 1 \implies y^2 + \frac{1}{16} = 1 \implies y^2 = 1 - \frac{1}{16} = \frac{15}{16} \]
Thus:
\[ \sin(b) = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4} \]

### Step 5: Substitute back into the sine addition formula
Now substituting back into the formula:
\[ \sin\left(\frac{\pi}{6} + \cos^{-1}\left(\frac{1}{4}\right)\right) = \left(\frac{1}{2}\right)\left(\frac{1}{4}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{15}}{4}\right) \]
Calculating each term:
\[ = \frac{1}{8} + \frac{\sqrt{3} \cdot \sqrt{15}}{8} = \frac{1}{8} + \frac{\sqrt{45}}{8} = \frac{1 + 3\sqrt{5}}{8} \]

### Final Answer
Thus, the evaluated expression is:
\[ \sin\left(\frac{\pi}{6} + \cos^{-1}\left(\frac{1}{4}\right)\right) = \frac{1 + 3\sqrt{5}}{8} \]