\[ 2 - 2\sin^2 \theta - 3\sin \theta = 0 \]

["Solving the Trigonometric Equation: 2 – 2 sin²θ – 3 sin θ = 0", "Trigonometry is a fundamental part of mathematics, appearing in physics, engineering, and various applied sciences. One common challenge is solving trigonometric equations involving sine functions, especially nonlinear ones. This article dives deep into solving the equation:", "### ( 2 - 2 \sin^2 \ heta - 3 \sin \ heta = 0 )", "#### Understanding the Equation", "The given equation is a quadratic in terms of ( \sin \ heta ). Let’s simplify it by using a substitution to make it more manageable.", "Let ( x = \sin \ heta ). Substituting, the equation becomes:\n[\n2 - 2x^2 - 3x = 0\n]\nRewriting it in standard quadratic form:\n[\n-2x^2 - 3x + 2 = 0\n]\nOr equivalently:\n[\n2x^2 + 3x - 2 = 0\n]", "#### Solving the Quadratic Equation", "We solve the quadratic equation ( 2x^2 + 3x - 2 = 0 ) using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere ( a = 2 ), ( b = 3 ), and ( c = -2 ).", "Calculate the discriminant:\n[\n\Delta = b^2 - 4ac = 3^2 - 4(2)(-2) = 9 + 16 = 25\n]", "Now substitute into the formula:\n[\nx = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}\n]", "This gives two solutions:\n[\nx = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}\n]\n[\nx = \frac{-3 - 5}{4} = \frac{-8}{4} = -2\n]", "#### Restricting the Solutions to Valid Sine Values", "Since ( x = \sin \ heta ), and the sine function only takes values in the interval ([-1, 1]), we discard ( x = -2 ) as it is outside this range.", "Thus, the only valid solution is:\n[\n\sin \ heta = \frac{1}{2}\n]", "#### Solving for θ", "We now solve:\n[\n\sin \ heta = \frac{1}{2}\n]", "Within the principal range ( [0^\circ, 360^\circ) ), the solutions are:\n[\n\ heta = 30^\circ \quad \ ext{and} \quad \ heta = 150^\circ\n]", "These are the standard angles where sine equals ( \frac{1}{2} ). Since sine is periodic with period ( 360^\circ ), the general solution includes all angles:\n[\n\ heta = 30^\circ + 360^\circ n \quad \ ext{and} \quad \ heta = 150^\circ + 360^\circ n \quad \ ext{for any integer } n\n]", "#### Summary", "The equation ( 2 - 2\sin^2 \ heta - 3\sin \ heta = 0 ) simplifies to a quadratic in ( \sin \ heta ), yielding the key solution:\n[\n\sin \ heta = \frac{1}{2}\n]\nWhich corresponds to angles:\n[\n\ heta = 30^\circ + 360^\circ n \quad \ ext{and} \quad \ heta = 150^\circ + 360^\circ n\n]", "---", "### Why This Equation Matters", "This type of equation appears frequently in wave mechanics, electrical engineering (AC circuits), and oscillatory systems. Mastering trigonometric identities and substitution techniques enables efficient solutions that are essential for applying trigonometry in real-world problems.", "#### Practice Problem", "Solve ( 2 - 2\sin^2 \ heta - 3\sin \ heta = 0 ) using trigonometric identities:\n1. Substitute ( x = \sin \ heta ).\n2. Rewrite and solve the quadratic.\n3. Restrict solutions to valid sine values.\n4. Convert ( \sin \ heta = \frac{1}{2} ) to general angle solutions.", "---", "Key Takeaways\n- Always substitute to reduce nonlinear trigonometric equations to algebraic form.\n- Discard solutions outside the domain of sine.\n- Use the unit circle to express all solutions.", "Mastering these steps strengthens your trigonometric problem-solving skills—play a vital role in both academic work and technical applications.", "---", "For further exploration, consult trigonometric sum and product identities, or use graphing tools to visualize the solutions."]