Find `"a"` and `"b"` such that `a lt 3 "cos" theta+5 "sin"(theta-(pi)/(6)) lt b`, holds good for all values of `theta`.

Answer

To solve the inequality \( a < 3 \cos \theta + 5 \sin\left(\theta - \frac{\pi}{6}\right) < b \) for all values of \( \theta \), we will first rewrite the expression \( 5 \sin\left(\theta - \frac{\pi}{6}\right) \) using the sine subtraction formula.

### Step 1: Rewrite the sine term
Using the sine subtraction formula:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
we can rewrite \( 5 \sin\left(\theta - \frac{\pi}{6}\right) \) as:
\[ 5 \left( \sin \theta \cos\left(\frac{\pi}{6}\right) - \cos \theta \sin\left(\frac{\pi}{6}\right) \right) \]

### Step 2: Substitute the values of cosine and sine
We know:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Substituting these values gives:
\[ 5 \sin\left(\theta - \frac{\pi}{6}\right) = 5 \left( \sin \theta \cdot \frac{\sqrt{3}}{2} - \cos \theta \cdot \frac{1}{2} \right) = \frac{5\sqrt{3}}{2} \sin \theta - \frac{5}{2} \cos \theta \]

### Step 3: Combine terms
Now we can express the entire inequality:
\[ 3 \cos \theta + 5 \sin\left(\theta - \frac{\pi}{6}\right) = 3 \cos \theta + \left(\frac{5\sqrt{3}}{2} \sin \theta - \frac{5}{2} \cos \theta\right) \]
This simplifies to:
\[ \left(3 - \frac{5}{2}\right) \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta = \frac{1}{2} \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta \]

### Step 4: Find the maximum and minimum values
To find the range of the expression \( \frac{1}{2} \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta \), we can use the formula:
\[ R = \sqrt{A^2 + B^2} \]
where \( A = \frac{1}{2} \) and \( B = \frac{5\sqrt{3}}{2} \):
\[ R = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{5\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{75}{4}} = \sqrt{19} \]

### Step 5: Determine the bounds
The expression \( \frac{1}{2} \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta \) will range from \( -\sqrt{19} \) to \( \sqrt{19} \). Therefore, we have:
\[ -\sqrt{19} < \frac{1}{2} \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta < \sqrt{19} \]

### Conclusion
Thus, the values of \( a \) and \( b \) are:
\[ a = -\sqrt{19}, \quad b = \sqrt{19} \]