Verified Answer :

  • Simplify the equation using the sum-to-product identity: \(\sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\).
  • Substitute \(A = x + \frac{\pi}{3}\) and \(B = x - \frac{\pi}{3}\) to reduce the equation to \(\sin(x) = 1\).
  • Solve \(\sin(x) = 1\) to find the general solution \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\).
  • The final set of solutions is \(\boxed{x = \frac{\pi}{2} + 2k\pi}\).

Explanation

  1. Analysis of the Problem

The objective is to solve the trigonometric equation \(\sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=1\) for \(x\). This equation involves the sum of two sine functions with angles shifted by \(\pm \frac{\pi}{3}\). We can use trigonometric identities to simplify this expression. Specifically, the sum-to-product identity \(\sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\) is highly effective here. Alternatively, we can use the angle addition and subtraction formulas for sine.

  1. Simplifying the Equation

Let \(A = x + \frac{\pi}{3}\) and \(B = x - \frac{\pi}{3}\). Using the sum-to-product identity \(\sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\):

  1. Compute the average angle: \(\frac{A+B}{2} = \frac{(x + \frac{\pi}{3}) + (x - \frac{\pi}{3})}{2} = \frac{2x}{2} = x\).
  2. Compute the half-difference of the angles: \(\frac{A-B}{2} = \frac{(x + \frac{\pi}{3}) - (x - \frac{\pi}{3})}{2} = \frac{\frac{2\pi}{3}}{2} = \frac{\pi}{3}\).

Substituting these into the identity, the left-hand side becomes \(2 \sin(x) \cos\left(\frac{\pi}{3}\right)\). Since \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), the expression simplifies to \(2 \sin(x) \times \left(\frac{1}{2}\right) = \sin(x)\). Thus, the original equation \(\sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=1\) simplifies to \(\sin(x) = 1\).

  1. Solving the Simplified Equation

We now have the equation \(\sin(x) = 1\). The sine function equals \(1\) when its argument \(x\) is of the form \(\frac{\pi}{2} + 2k\pi\), where \(k\) is any integer. Thus, the general solution for \(x\) is \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\).

  1. Final Conclusion

The solutions to the trigonometric equation \(\sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=1\) are given by \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\). Representing this set, the values of \(x\) are \(\boxed{x = \frac{\pi}{2} + 2k\pi}\) for any integer \(k\).

Examples

Trigonometric equations like this frequently appear in physics and engineering, particularly when analyzing wave interference. For example, when two sound waves or electromagnetic waves overlap (superposition), the resulting wave's amplitude is the sum of the individual amplitudes. Solving for when the total amplitude reaches a specific value (like the maximum) helps engineers determine conditions for constructive interference, which is crucial for signal optimization in communication systems or acoustic design in concert halls.

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