Question:
The maximum value of \(\sin x+\sin(x+1)\) is \(k\cos\frac12\). Find \(k\).
The maximum value of \(\sin x+\sin(x+1)\) is \(k\cos\frac12\). Find \(k\).
Show Hint
Whenever a sum of sine functions appears, first convert it into product form using sum-to-product identities.
Updated On: Jun 8, 2026
- 1
- 3
- 2
- None of these
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The Correct Option is C
Solution and Explanation
Concept:
Use the identity
\[
\sin A+\sin B
=
2\sin\frac{A+B}{2}
\cos\frac{A-B}{2}
\]
to combine two sine functions.
Step 1: Apply sum-to-product identity.
\[ \sin x+\sin(x+1) = 2\sin\left(x+\frac12\right) \cos\frac12 \]
Step 2: Find maximum value.
Since \[ -1\le \sin\left(x+\frac12\right)\le1, \] maximum occurs when \[ \sin\left(x+\frac12\right)=1. \] Therefore \[ \max= 2\cos\frac12 \] Comparing with \[ k\cos\frac12, \] we obtain \[ k=2. \] \[ \boxed{k=2} \]
Step 1: Apply sum-to-product identity.
\[ \sin x+\sin(x+1) = 2\sin\left(x+\frac12\right) \cos\frac12 \]
Step 2: Find maximum value.
Since \[ -1\le \sin\left(x+\frac12\right)\le1, \] maximum occurs when \[ \sin\left(x+\frac12\right)=1. \] Therefore \[ \max= 2\cos\frac12 \] Comparing with \[ k\cos\frac12, \] we obtain \[ k=2. \] \[ \boxed{k=2} \]
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