Find all angles \( \theta \in [0^\circ, 360^\circ] \) that satisfy the equation \(\sin(2\theta) = \cos(\theta)\).

["# All Angles ( \ heta \in [0^\circ, 360^\circ] ) That Satisfy ( \sin(2\ heta) = \cos(\ heta) )", "Solving trigonometric equations often reveals important relationships between sine and cosine functions. One particularly insightful identity is finding all angles ( \ heta ) in the interval ([0^\circ, 360^\circ]) that satisfy the equation:", "[\n\sin(2\ heta) = \cos(\ heta)\n]", "In this article, we explore step-by-step how to find all solutions in the specified interval, leveraging key trigonometric identities and algebraic techniques.", "---", "## Understanding the Equation", "We start with:", "[\n\sin(2\ heta) = \cos(\ heta)\n]", "Recall the double angle identity:", "[\n\sin(2\ heta) = 2\sin(\ heta)\cos(\ heta)\n]", "Substituting this identity into the equation gives:", "[\n2\sin(\ heta)\cos(\ heta) = \cos(\ heta)\n]", "---", "## Simplifying the Equation", "Bring all terms to one side:", "[\n2\sin(\ heta)\cos(\ heta) - \cos(\ heta) = 0\n]", "Factor out ( \cos(\ heta) ):", "[\n\cos(\ heta)(2\sin(\ heta) - 1) = 0\n]", "This product equals zero when any factor is zero, giving two cases:", "1. ( \cos(\ heta) = 0 )\n2. ( 2\sin(\ heta) - 1 = 0 ) → ( \sin(\ heta) = \frac{1}{2} )", "We now solve each case over ( \ heta \in [0^\circ, 360^\circ] ).", "---", "## Case 1: Solve ( \cos(\ heta) = 0 )", "The cosine function equals zero at:", "[\n\ heta = 90^\circ \quad \ ext{and} \quad \ heta = 270^\circ\n]", "Both values lie within ([0^\circ, 360^\circ]).\nSo, solutions from this case:", "[\n\ heta = 90^\circ, 270^\circ\n]", "---", "## Case 2: Solve ( \sin(\ heta) = \frac{1}{2} )", "We now find all ( \ heta \in [0^\circ, 360^\circ] ) such that sine equals ( \frac{1}{2} ).", "The sine function equals ( \frac{1}{2} ) at:", "[\n\ heta = 30^\circ \quad \ ext{and} \quad \ heta = 150^\circ\n]", "(These angles lie in Quadrants I and II, where sine is positive.)", "So, solutions from this case:", "[\n\ heta = 30^\circ, 150^\circ\n]", "---", "## Final List of Solutions", "Combining both cases, the complete set of angles ( \ heta ) that satisfy ( \sin(2\ heta) = \cos(\ heta) ) in ([0^\circ, 360^\circ]) is:", "[\n\ heta = 30^\circ, 90^\circ, 150^\circ, 270^\circ\n]", "---", "## Verification with Key Identities", "Let’s verify each solution in the original equation:", "- For ( \ heta = 30^\circ ):\n ( \sin(60^\circ) = \frac{\sqrt{3}}{2} ), ( \cos(30^\circ) = \frac{\sqrt{3}}{2} ) → valid.", "- For ( \ heta = 90^\circ ):\n ( \sin(180^\circ) = 0 ), ( \cos(90^\circ) = 0 ) → valid.", "- For ( \ heta = 150^\circ ):\n ( \sin(300^\circ) = -\frac{\sqrt{3}}{2} ), ( \cos(150^\circ) = -\frac{\sqrt{3}}{2} ) → valid.", "- For ( \ heta = 270^\circ ):\n ( \sin(540^\circ) = \sin(180^\circ) = 0 ), ( \cos(270^\circ) = 0 ) → valid.", "All solutions are confirmed correct.", "---", "## Why This Equation Matters", "Understanding where ( \sin(2\ heta) = \cos(\ heta) ) holds helps in modeling periodic phenomena—such as wave interference, alternating current, or circular motion—where phase differences between sine and cosine functions are critical.", "---", "## Summary", "To solve ( \sin(2\ heta) = \cos(\ heta) ) for ( \ heta \in [0^\circ, 360^\circ] ):", "- Use the identity ( \sin(2\ heta) = 2\sin(\ heta)\cos(\ heta) ),", "- Rearrange into factorized form: ( \cos(\ heta)(2\sin(\ heta) - 1) = 0 ),", "- Solve ( \cos(\ heta) = 0 ) → solutions ( 90^\circ, 270^\circ ),", "- Solve ( \sin(\ heta) = \frac{1}{2} ) → solutions ( 30^\circ, 150^\circ ).", "All solutions:", "[\n\boxed{ \ heta = 30^\circ,\ 90^\circ,\ 150^\circ,\ 270^\circ }\n]", "---", "## SEO Optimization Tips", "- Target keywords: solve sin(2θ) = cosθ, find angles where sin(2θ) = cosθ, solve trigonometric equation 2θ = cosθ\n- Use long-tail phrases: “angle solutions for sin(2x) = cos(x) in degrees”\n- Include structured format: Bullet points for solutions improve skimmability\n- Embed FAQs: “What angles satisfy sin(2θ) = cos(θ)?”\n- Link internally: To related articles on double angle identities and trigonometric equations", "---", "By mastering such equations, you gain powerful tools for analyzing oscillatory systems—a cornerstone of trigonometry, physics, and engineering."]