Maximum value of (2sin⁡θ+3cos⁡θ) is

Correct option is B

Solution

We are tasked to find the maximum value of $$2\sin\theta + 3\cos\theta$$​

Step-by-Step Solution:

1. Given Expression:

$$y = 2\sin\theta + 3\cos\theta$$​

2. Rewriting Using a Single Trigonometric Function:

Express y in the form $$R\sin(\theta + \phi)$$​ , where:

$$R = \sqrt{a^2 + b^2}, \quad \tan\phi = \frac{b}{a}$$​

Here, a = 2 and b = 3 .

Calculate R :

$$R = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$$​

The expression becomes:

$$y = R\sin(\theta + \phi) = \sqrt{13}\sin(\theta + \phi)$$​

3. Maximum Value of $$\sin(\theta + \phi)$$​ :

The maximum value of $$\sin(\theta + \phi)$$​ is 1. Substituting this maximum value:

$$y_{\text{max}} = \sqrt{13} \cdot 1 = \sqrt{13}$$​

Final Answer:

$$\boxed{\sqrt{13}}$$​