Question: Find all angles $z \in [0^\circ, 360^\circ]$ satisfying $2\sin^2 z - 3\sin z + 1 = 0$.

["Title: Solve the Trigonometric Equation: Find All Angles $z \in [0^\circ, 360^\circ]$ Satisfying $2\sin^2 z - 3\sin z + 1 = 0$", "---", "Introduction\nUnderstanding trigonometric equations is essential for mastering both theoretical and applied mathematics. One frequently encountered problem involves solving polynomial forms in sine and cosine. In this article, we’ll explore how to find all angles $ z $ in degrees between $ 0^\circ $ and $ 360^\circ $ that satisfy the equation:", "$$\n2\sin^2 z - 3\sin z + 1 = 0\n$$", "This is a quadratic equation in terms of $ \sin z $, and by transforming it cleverly, we can efficiently find all valid solutions within the desired interval.", "---", "Step 1: Substitute to Simplify the Equation\nLet’s let $ x = \sin z $. Then the given equation becomes a standard quadratic:", "$$\n2x^2 - 3x + 1 = 0\n$$", "We now solve this quadratic equation using factoring.", "---", "Step 2: Factor the Quadratic\nWe look for two numbers that multiply to $ 2 \ imes 1 = 2 $ and add to $ -3 $. These numbers are $ -1 $ and $ -2 $:", "$$\n2x^2 - 2x - x + 1 = 0\n\Rightarrow 2x(x - 1) -1(x - 1) = 0\n\Rightarrow (2x - 1)(x - 1) = 0\n$$", "---", "Step 3: Solve for $ x = \sin z $\nSet each factor equal to zero:", "- $ 2x - 1 = 0 \Rightarrow x = \frac{1}{2} $\n- $ x - 1 = 0 \Rightarrow x = 1 $", "So the possible values of $ \sin z $ are:\n$$\n\sin z = \frac{1}{2} \quad \ ext{and} \quad \sin z = 1\n$$", "---", "Step 4: Find All $ z \in [0^\circ, 360^\circ] $ for Each Case", "### Case 1: $ \sin z = \frac{1}{2} $\nThe sine function equals $ \frac{1}{2} $ at two standard angles in the interval $ [0^\circ, 360^\circ] $:\n- $ z = 30^\circ $ (first quadrant)\n- $ z = 150^\circ $ (second quadrant)", "### Case 2: $ \sin z = 1 $\nThe sine function reaches 1 at:\n- $ z = 90^\circ $", "---", "Step 5: List All Valid Solutions\nCombining both cases, all angles $ z $ satisfying the equation are:\n$$\nz = 30^\circ, \quad z = 90^\circ, \quad z = 150^\circ\n$$", "---", "Conclusion\nSolving $ 2\sin^2 z - 3\sin z + 1 = 0 $ reduces to a simple quadratic in $ \sin z $, which yields two key solutions: $ \sin z = \frac{1}{2} $ and $ \sin z = 1 $. Within $ [0^\circ, 360^\circ] $, the solutions are $ 30^\circ, 90^\circ, $ and $ 150^\circ $. This method demonstrates how substitution and factoring simplify trigonometric equations, making them easier to solve efficiently.", "---", "Key Takeaways:\n- Always substitute to reduce higher trig functions to polynomials when possible.\n- Factor carefully and apply the zero-product property.\n- Recall the exact values and reference angles for sine and cosine.\n- Always verify solutions fall within the specified interval.", "Mastering these steps builds strong problem-solving skills useful in physics, engineering, and advanced math.", "---", "Keywords:\nangles $ z \in [0^\circ, 360^\circ] $, solve $ 2\sin^2 z - 3\sin z + 1 = 0 $, trigonometric equation, sine identity, quadratic in $ \sin z $, solve $ \sin z = \frac{1}{2} $, solve $ \sin z = 1 $, trigonometric solutions, angle solving, interval solutions, exact values, reference angles.", "---", "Meta Description:\nFind all angles $ z $ in $ [0^\circ, 360^\circ] $ satisfying $ 2\sin^2 z - 3\sin z + 1 = 0 $. Clear step-by-step solution for students and math enthusiasts."]