Question

The maximum value of $\sin(x) + \sin(x + 1)$ is $k \cos^{\frac{1}{2}}$ Then the value of $k$ is:
 

• A. 1
• B. 2
• C. None of these
• D. 3

Hint:

When dealing with trigonometric identities, use sum-to-product identities to simplify and find maximum values.

Answer 1

The Correct Option is B

We are given the expression $\sin(x) + \sin(x + 1)$. Using the sum identity for sine, we have: \[ \sin(x) + \sin(x + 1) = 2 \sin\left(\frac{2x + 1}{2}\right) \cos\left(\frac{1}{2}\right) \] To maximize this expression, we need to maximize both sine and cosine functions. The maximum value of sine and cosine is 1. Hence, the maximum value of the given expression is: \[ 2 \times 1 \times 1 = 2 \] Thus, $k = 2$.