Find the maximum value of 3cos θ + 4sin θ + 5cos(θ + π/6).
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Step-by-Step Solution
Step 1: Expand the expression
First, we expand the term 5cos(θ+π/6) using the trigonometric identity cos(A+B)=cosAcosB−sinAsinB. This will allow us to combine like terms involving cosθ and sinθ.
Step 2: Substitute known values and simplify
We substitute the known values for cos(π/6)=3/2 and sin(π/6)=1/2. Then, we group the coefficients of cosθ and sinθ to simplify the expression into the form Acosθ+Bsinθ.
Step 3: Apply the maximum value formula
For an expression of the form Acosθ+Bsinθ, its maximum value is given by A2+B2. Here, A=26+53 and B=23.
Step 4: Calculate A2
We calculate the square of A. Remember to square both the numerator and the denominator, and use the formula (a+b)2=a2+2ab+b2 for the numerator.
Step 5: Calculate B2 and A2+B2
Next, we calculate the square of B. Then, we add A2 and B2 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 A2+B2 to find the maximum value of the given expression.