السؤال: أوجد جميع الزوايا $z \in [0^\circ, 360^\circ]$ التي تحقق المعادلة $2\sin z + \sqrt{3} = 0$.

["Optimized SEO Article: Find All Angles ( z \in [0^\circ, 360^\circ] ) That Satisfy ( 2\sin z + \sqrt{3} = 0 )", "If you're solving trigonometric equations, understanding how to find all solutions within a specific interval is essential. One common problem is solving the equation:", "[\n2\sin z + \sqrt{3} = 0\n]", "for angles ( z ) in degrees between ( 0^\circ ) and ( 360^\circ ). This article guides you step-by-step through solving this equation and identifies all valid solutions in the target range.", "---", "### Understanding the Equation", "Start by isolating ( \sin z ):", "[\n2\sin z = -\sqrt{3}\n]\n[\n\sin z = -\frac{\sqrt{3}}{2}\n]", "This means we need to find all angles ( z ) in degrees between ( 0^\circ ) and ( 360^\circ ) where the sine value is ( -\frac{\sqrt{3}}{2} ).", "---", "### Quick Recap: Where is ( \sin z = -\frac{\sqrt{3}}{2} )?", "The reference angles where ( \sin \ heta = \frac{\sqrt{3}}{2} ) are ( 60^\circ ) and ( 120^\circ ) in the unit circle. Since sine is negative in the third and fourth quadrants:", "- In the third quadrant: ( z = 180^\circ + 60^\circ = 240^\circ )\n- In the fourth quadrant: ( z = 360^\circ - 60^\circ = 300^\circ )", "These are the primary solutions in the interval ( [0^\circ, 360^\circ] ).", "---", "### Verify Solutions Are Within the Required Interval", "Both ( 240^\circ ) and ( 300^\circ ) clearly fall within ( [0^\circ, 360^\circ] ), so they are valid.", "---", "### Final Answer", "The complete set of solutions in degrees satisfying:", "[\n2\sin z + \sqrt{3} = 0\n]", "is:", "[\n\boxed{z = 240^\circ \quad \ ext{and} \quad z = 300^\circ}\n]", "---", "### Tips for Mastering Trigonometric Solutions", "- Always isolate the trigonometric function first.\n- Remember where sine is positive/negative: 1st & 2nd (positive), 3rd & 4th (negative).\n- Use reference angles from the unit circle.\n- Check all solutions lie in the specified interval ( [0^\circ, 360^\circ] ).\n- Use symmetry and periodicity to verify completeness.", "---", "If you’re studying angles, mastering sine values and their quadrants will help you solve more complex trig equations quickly and confidently. Whether for homework, exams, or practical applications, knowing how to find all solutions in a range is key — and mastering this equation is a great step forward!"]