السؤال: أوجد جميع الزوايا $z \in [0^\circ, 360^\circ]$ التي تحقق المعادلة $\cos 2z = \sin z$.

["Title: How to Solve $\cos 2z = \sin z$: Find All Solutions in $[0^\circ, 360^\circ]$", "Navigating trigonometric equations can feel challenging, but with the right steps, solving $\cos 2z = \sin z$ becomes manageable. In this SEO-optimized article, we break down everything you need to know to find all angles $z \in [0^\circ, 360^\circ]$ that satisfy this equation.", "---", "### Understanding the Equation: $\cos 2z = \sin z$", "The equation $\cos 2z = \sin z$ combines double-angle and sine functions. To solve it, we use trigonometric identities to rewrite both sides in comparable forms.", "Recall the double-angle identity:\n$$\n\cos 2z = 1 - 2\sin^2 z\n$$\nThis converts the left-hand side into a sine-only expression, allowing substitution.", "So the equation becomes:\n$$\n1 - 2\sin^2 z = \sin z\n$$", "---", "### Rewriting as a Quadratic Equation", "Rearranging terms gives a quadratic in $\sin z$:\n$$\n2\sin^2 z + \sin z - 1 = 0\n$$", "Let $x = \sin z$. Then:\n$$\n2x^2 + x - 1 = 0\n$$", "Use the quadratic formula:\n$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 + 8}}{4} = \frac{-1 \pm 3}{4}\n$$", "This yields two solutions:\n$$\nx = \frac{1}{2}, \quad x = -1\n$$", "So $\sin z = \frac{1}{2}$ or $\sin z = -1$", "---", "### Solving $\sin z = \frac{1}{2}$ in $[0^\circ, 360^\circ]$", "We recall the angles where sine is $\frac{1}{2}$:\n$$\nz = 30^\circ, \quad z = 150^\circ\n$$\nThese lie within the interval.", "---", "### Solving $\sin z = -1$ in $[0^\circ, 360^\circ]$", "$\sin z = -1$ occurs at:\n$$\nz = 270^\circ\n$$\nThis is the only solution in the given range.", "---", "### Collecting All Solutions", "From above, the solutions are:\n$$\nz = 30^\circ, \quad 150^\circ, \quad 270^\circ\n$$", "These are all distinct and lie within $[0^\circ, 360^\circ]$.", "---", "### Verification with Original Equation", "Let’s double-check each value:", "- $z = 30^\circ$:\n $\cos(60^\circ) = 0.5$, $\sin(30^\circ) = 0.5$ → equal ✔️", "- $z = 150^\circ$:\n $\cos(300^\circ) = \cos(-60^\circ) = 0.5$, $\sin(150^\circ) = 0.5$ → equal ✔️", "- $z = 270^\circ$:\n $\cos(540^\circ) = \cos(180^\circ) = -1$, $\sin(270^\circ) = -1$ → equal ✔️", "All satisfy the equation.", "---", "### Conclusion: Final Answer", "All angles $z$ in degrees satisfying $\cos 2z = \sin z$ within $[0^\circ, 360^\circ]$ are:", "$$\n\boxed{30^\circ,\ 150^\circ,\ 270^\circ}\n$$", "Using precise trigonometric identities, algebraic manipulation, and careful verification ensures accuracy. This approach exemplifies effective problem-solving strategies valuable for learners and practitioners seeking clarity in trigonometry.", "---", "### Key SEO Keywords for Search Optimization:\n- Solve $\cos 2z = \sin z$\n- Find all $z$ in $[0^\circ, 360^\circ]$ satisfying $\cos 2z = \sin z$\n- Trigonometric equations solution\n- $\sin z = \frac{1}{2}$ and $\sin z = -1$\n- Step-by-step trigonometry tutorial\n- Finding angles $\cos 2z = \sin z$ in degrees", "By targeting these keywords and employing a structured, editorial-style explanation, this article ranks well and provides practical, shareable knowledge."]