If `ale 3 cos x + 5 sin ( x - pi //6) leb` for all x, then (a,b ) =

Options

A. `( - sqrt( 19), sqrt( 19))`
B. `( - 17,17)`
C. `( - sqrt( 21), sqrt( 21)`
D. none

Answer

To solve the problem, we need to find the values of \(a\) and \(b\) such that:

\[ a \leq 3 \cos x + 5 \sin\left(x - \frac{\pi}{6}\right) \leq b \]

for all \(x\). We will follow these steps:

### Step 1: Rewrite the expression

We start with the expression \(3 \cos x + 5 \sin\left(x - \frac{\pi}{6}\right)\). We can use the sine subtraction formula:

\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]

Thus, we rewrite \(5 \sin\left(x - \frac{\pi}{6}\right)\):

\[ 5 \sin\left(x - \frac{\pi}{6}\right) = 5 \left(\sin x \cos\frac{\pi}{6} - \cos x \sin\frac{\pi}{6}\right) \]

Using \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin\frac{\pi}{6} = \frac{1}{2}\):

\[ 5 \sin\left(x - \frac{\pi}{6}\right) = 5 \left(\sin x \cdot \frac{\sqrt{3}}{2} - \cos x \cdot \frac{1}{2}\right) = \frac{5\sqrt{3}}{2} \sin x - \frac{5}{2} \cos x \]

Now, substituting this back into the original expression:

\[ 3 \cos x + 5 \sin\left(x - \frac{\pi}{6}\right) = 3 \cos x + \frac{5\sqrt{3}}{2} \sin x - \frac{5}{2} \cos x \]

### Step 2: Combine like terms

Combining the cosine terms:

\[ \left(3 - \frac{5}{2}\right) \cos x + \frac{5\sqrt{3}}{2} \sin x = \frac{1}{2} \cos x + \frac{5\sqrt{3}}{2} \sin x \]

### Step 3: Find the maximum and minimum values

We can express the combined terms in the form \(p \cos x + q \sin x\), where \(p = \frac{1}{2}\) and \(q = \frac{5\sqrt{3}}{2}\).

The maximum and minimum values of \(p \cos x + q \sin x\) can be found using the following formulas:

\[ \text{Maximum} = \sqrt{p^2 + q^2} \]
\[ \text{Minimum} = -\sqrt{p^2 + q^2} \]

Calculating \(p^2 + q^2\):

\[ p^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ q^2 = \left(\frac{5\sqrt{3}}{2}\right)^2 = \frac{75}{4} \]

Thus:

\[ p^2 + q^2 = \frac{1}{4} + \frac{75}{4} = \frac{76}{4} = 19 \]

### Step 4: Calculate the maximum and minimum values

Now, we find the maximum and minimum:

\[ \text{Maximum} = \sqrt{19} \]
\[ \text{Minimum} = -\sqrt{19} \]

### Conclusion

Thus, the values of \(a\) and \(b\) are:

\[ (a, b) = (-\sqrt{19}, \sqrt{19}) \]