To solve the equation [tex]\(\cos \left(\frac{\pi}{6} + x\right) + \sin \left(\frac{\pi}{3} + x\right) = 0\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex], let's work through the problem step-by-step using trigonometric identities and simplifications.
1. Start by applying the angle addition formulas for cosine and sine:
- [tex]\(\cos \left(\frac{\pi}{6} + x\right) = \cos\left(\frac{\pi}{6}\right)\cos(x) - \sin\left(\frac{\pi}{6}\right)\sin(x)\)[/tex]
- [tex]\(\sin \left(\frac{\pi}{3} + x\right) = \sin\left(\frac{\pi}{3}\right)\cos(x) + \cos\left(\frac{\pi}{3}\right)\sin(x)\)[/tex]
2. Substitute the known values for [tex]\(\cos\left(\frac{\pi}{6}\right)\)[/tex], [tex]\(\sin\left(\frac{\pi}{6}\right)\)[/tex], [tex]\(\sin\left(\frac{\pi}{3}\right)\)[/tex], and [tex]\(\cos\left(\frac{\pi}{3}\right)\)[/tex]:
- [tex]\(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)[/tex]
3. Substitute these values into the original equation:
[tex]\[
\left(\frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)\right) + \left(\frac{\sqrt{3}}{2}\cos(x) + \frac{1}{2}\sin(x)\right) = 0
\][/tex]
4. Combine the like terms:
[tex]\[
\frac{\sqrt{3}}{2}\cos(x) + \frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x) + \frac{1}{2}\sin(x) = 0 \implies \sqrt{3}\cos(x) = 0
\][/tex]
5. Simplify:
[tex]\[
\cos(x) = 0
\][/tex]
6. Determine the solutions for [tex]\(\cos(x) = 0\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex]:
- [tex]\(\cos(x) = 0\)[/tex] when [tex]\(x = \frac{\pi}{2}, \frac{3\pi}{2}\)[/tex]
Thus, the solutions to the equation [tex]\(\cos \left(\frac{\pi}{6} + x\right) + \sin \left(\frac{\pi}{3} + x\right) = 0\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\boxed{\frac{\pi}{2}, \frac{3\pi}{2}}
\][/tex]
