Question:
The maximum value of $3 \sin x + 4 \cos x + 5$ is:
The maximum value of $3 \sin x + 4 \cos x + 5$ is:
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$(3, 4, 5)$ is a standard Pythagorean triple, so $\sqrt{3^2+4^2}$ is instantly $5$. Add $5$ to get $10$.
Updated On: May 31, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Concept
The expression $a \sin x + b \cos x$ always lies in the range $[-\sqrt{a^2 + b^2}, \sqrt{a^2 + b^2}]$.
Step 2: Meaning
Therefore, the expression $a \sin x + b \cos x + c$ has a maximum value of $c + \sqrt{a^2 + b^2}$.
Step 3: Analysis
Identify the constants from the given expression: $a = 3$, $b = 4$, and $c = 5$. Compute the maximum value: \[ \text{Maximum Value} = 5 + \sqrt{3^2 + 4^2} = 5 + \sqrt{9 + 16} = 5 + \sqrt{25} = 5 + 5 = 10 \]
Step 4: Conclusion
The maximum value that the function can achieve is $10$. Final Answer: (B)
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