Question

The maximum value of 3 cos\(x\) \(+\) 4 sin\(x\) \(+\) 8 is

• A. 15
• B. 3
• C. 13
• D. 7

Answer 1

The Correct Option is C

To find the maximum value of the expression \(3 \cos(x) + 4 \sin(x) + 8\), we can use the concept of the amplitude of a linear combination of sine and cosine functions. 

1. First, identify the coefficients of the sine and cosine functions:
   • For \(\cos(x)\), the coefficient is 3.
   • For \(\sin(x)\), the coefficient is 4.
2. Compute the amplitude of the combination of the sine and cosine functions using the formula: \(R = \sqrt{a^2 + b^2}\), where \(a\) is the coefficient of \(\cos(x)\) and \(b\) is the coefficient of \(\sin(x)\).
   • Here, \(a = 3\) and \(b = 4\).
   • Therefore, \(R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
3. The maximum value of the expression \(3 \cos(x) + 4 \sin(x)\) is equal to the amplitude, \(R = 5\).
4. Add the constant term from the original expression to the maximum amplitude to find the maximum of the whole expression \(3 \cos(x) + 4 \sin(x) + 8\).
   • The maximum value is \(5 + 8 = 13\).
5. Therefore, the maximum value of the expression is \(13\).

Hence, the correct answer is 13.