Find the maximum value of 3cos θ + 4sin θ + 5cos(θ + π/6).

Step 1: Expand the expression

First, we expand the term $5\cos(\theta + \pi/6)$ using the trigonometric identity $\cos(A+B) = \cos A \cos B - \sin A \sin B$. This will allow us to combine like terms involving $\cos \theta$ and $\sin \theta$.

Step 2: Substitute known values and simplify

We substitute the known values for $\cos(\pi/6) = \sqrt{3}/2$ and $\sin(\pi/6) = 1/2$. Then, we group the coefficients of $\cos \theta$ and $\sin \theta$ to simplify the expression into the form $A\cos \theta + B\sin \theta$.

Step 3: Apply the maximum value formula

For an expression of the form $A\cos \theta + B\sin \theta$, its maximum value is given by $\sqrt{A^2 + B^2}$. Here, $A = \frac{6+5\sqrt{3}}{2}$ and $B = \frac{3}{2}$.

Step 4: Calculate $A^2$

We calculate the square of $A$. Remember to square both the numerator and the denominator, and use the formula $(a+b)^2 = a^2 + 2ab + b^2$ for the numerator.

Step 5: Calculate $B^2$ and $A^2 + B^2$

Next, we calculate the square of $B$. Then, we add $A^2$ and $B^2$ together. Since they have a common denominator, we can simply add the numerators.

Step 6: Find the maximum value

Finally, we take the square root of the sum $A^2 + B^2$ to find the maximum value of the given expression.

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