Trigonometric Ratios and Identities
Maximum and Minimum Values
JEE Main 2019
Grade 11
Question:
The maximum value of $3 \cos \theta + 5 \sin(\theta - \frac{\pi}{6})$ for any real $\theta$ is:
(1) $\sqrt{19}$
(2) $\sqrt{34}$
(3) $\frac{\sqrt{19}}{2}$
(4) $\sqrt{21}$
Step-by-Step Solution
Key Concept: Expand $5 \sin(\theta - \pi/6)$ using the subtraction formula. Combine with $3 \cos \theta$ into the form $R \cos(\theta - \phi)$ or $R \sin(\theta + \phi)$. The maximum is $R$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)