A bioinformatician is analyzing a genomic sequence pattern and comes across the trigonometric equation \(\sin(2z) = \cos(z)\). Determine all angles \(z \in [0^\circ, 360^\circ]\) that satisfy this equation.

["Science Meets Math: Solving the Trigonometric Equation (\sin(2z) = \cos(z)) for Angles in [0°, 360°]", "In the fascinating intersection of biology and mathematics, bioinformaticians often encounter not only genetic sequences but also powerful mathematical tools to analyze patterns — including trigonometry. One intriguing challenge arises when analyzing periodic signals embedded in biological data, such as gene expression cycles or molecular oscillations, where equations like (\sin(2z) = \cos(z)) emerge naturally.", "This article explores how to analyze the trigonometric equation (\sin(2z) = \cos(z)) and determines all solutions within the interval ([0^\circ, 360^\circ]).", "---", "### Understanding the Equation: (\sin(2z) = \cos(z))", "The equation involves a double-angle sine function on the left and a cosine function on the right. Using trigonometric identities, we can rewrite (\sin(2z)) to simplify:", "[\n\sin(2z) = 2\sin(z)\cos(z)\n]", "So the equation becomes:", "[\n2\sin(z)\cos(z) = \cos(z)\n]", "---", "### Step 1: Rearranging the Equation", "Move all terms to one side:", "[\n2\sin(z)\cos(z) - \cos(z) = 0\n]", "Factor out (\cos(z)):", "[\n\cos(z)(2\sin(z) - 1) = 0\n]", "---", "### Step 2: Solving the Factored Equation", "Set each factor equal to zero:", "1. (\cos(z) = 0)\n2. (2\sin(z) - 1 = 0 \Rightarrow \sin(z) = \frac{1}{2})", "We solve each equation separately for (z \in [0^\circ, 360^\circ]).", "---", "### Solving (\cos(z) = 0)", "Cosine is zero at:", "[\nz = 90^\circ, \quad z = 270^\circ\n]", "These are the angles in the given interval where cosine vanishes.", "---", "### Solving (\sin(z) = \frac{1}{2})", "Sine equals (\frac{1}{2}) at:", "[\nz = 30^\circ, \quad z = 150^\circ\n]", "These are the standard reference angles in the first and second quadrants where sine is positive.", "---", "### Verifying All Solutions in ([0^\circ, 360^\circ])", "List all candidate solutions:\n[\nz = 30^\circ,\ 90^\circ,\ 150^\circ,\ 270^\circ\n]", "We confirm each satisfies the original equation:", "- For (z = 30^\circ):\n (\sin(60^\circ) = \frac{\sqrt{3}}{2},\ \cos(30^\circ) = \frac{\sqrt{3}}{2}) → equal.", "- For (z = 90^\circ):\n (\sin(180^\circ) = 0,\ \cos(90^\circ) = 0) → equal.", "- For (z = 150^\circ):\n (\sin(300^\circ) = -\frac{1}{2},\ \cos(150^\circ) = -\frac{\sqrt{3}}{2}) → not equal? Wait — correction needed.", "Wait:\n(\sin(2 \cdot 150^\circ) = \sin(300^\circ) = -\frac{1}{2})\n(\cos(150^\circ) = -\frac{\sqrt{3}}{2} \approx -0.866) → but (\sin(2z) = -\frac{1}{2} <br/>\ne -\frac{\sqrt{3}}{2}) — coincidence? No!", "Hold on: This suggests an error — let’s double-check.", "Wait: We factored (\cos(z)(2\sin(z) - 1) = 0), but did we preserve equivalence? Yes — algebraically valid.", "But let's test (z = 150^\circ):", "- (\sin(2z) = \sin(300^\circ) = -\frac{1}{2})\n- (\cos(z) = \cos(150^\circ) = -\frac{\sqrt{3}}{2} <br/>\ne -\frac{1}{2})", "So why does factoring suggest it’s a solution?", "Critical insight: Factoring is algebraically valid, but we must ensure no extraneous solutions are introduced — but here, the factoring assumes domain safety. However, the solutions (z = 150^\circ) and (270^\circ) do not satisfy the original equation?", "Wait — let’s re-analyze.", "For (z = 150^\circ):\n(\sin(2z) = \sin(300^\circ) = -\frac{1}{2})\n(\cos(150^\circ) = -\frac{\sqrt{3}}{2} \approx -0.866)", "But (-\frac{1}{2} <br/>\ne -0.866), so not a solution!", "So despite the factoring, (z = 150^\circ) does not satisfy (\sin(2z) = \cos(z)). Contradiction?", "Where is the flaw?", "Ah! The identity (\sin(2z) = 2\sin(z)\cos(z)) is correct. But when solving (2\sin(z)\cos(z) = \cos(z)), we factor out — but the equation is valid only when equating two expressions. However, our factorization is algebraically correct, so why discrepancy?", "Let’s recompute (\sin(2z)) at (z = 150^\circ):", "- (\sin(300^\circ) = \sin(180^\circ + 120^\circ) = -\sin(120^\circ) = -\frac{\sqrt{3}}{2}) — correct.\n- (\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}) — correct.", "So (\sin(2z) = -\frac{\sqrt{3}}{2},\ \cos(z) = -\frac{\sqrt{3}}{2}) → equal!", "But earlier I said (\sin(2z) = -\frac{1}{2}) — false. (\sin(300^\circ) = -\frac{\sqrt{3}}{2} \approx -0.866), not (-0.5). My arithmetic error.", "Correct value:\n(\sin(300^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}),\n(\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2})", "So both sides equal (-\frac{\sqrt{3}}{2}) → valid solution.", "Similarly, (z = 30^\circ):\n(\sin(60^\circ) = \frac{\sqrt{3}}{2},\ \cos(30^\circ) = \frac{\sqrt{3}}{2}) → equal.", "Now check (z = 270^\circ):\n(\sin(540^\circ) = \sin(180^\circ) = 0),\n(\cos(270^\circ) = 0) → equal.", "Thus, the solutions are:", "- (z = 30^\circ)\n- (z = 90^\circ)\n- (z = 150^\circ)\n- (z = 270^\circ)", "All satisfy the original equation.", "---", "### Final Step: Expressing the Solution Set", "The complete set of angles (z \in [0^\circ, 360^\circ]) that satisfy (\sin(2z) = \cos(z)) is:", "[\n\boxed{30^\circ,\ 90^\circ,\ 150^\circ,\ 270^\circ}\n]", "These solutions reflect periodic alignment between sine and cosine functions — a pattern that bioinformaticians might recognize in oscillatory biological data, such as circadian gene expression or protein concentration rhythms modeled using trigonometric functions.", "---", "### Conclusion", "The trigonometric equation (\sin(2z) = \cos(z)) reveals elegant symmetry when analyzed via algebraic identities and careful verification. By solving by factoring and checking, we uncover all valid angles within the domain — 30°, 90°, 150°, and 270° — exemplifying how mathematical reasoning enhances genomic and systems biology research.", "Whether modeling rhythmic gene expression or interpreting cyclical signals, tools like trigonometric equation solving remain indispensable.", "---", "Keywords: bioinformatician, genomic sequence analysis, trigonometric equation, (\sin(2z) = \cos(z)), solve for z, angles in [0°, 360°], mathematical biology, periodic functions, trigonometry in science."]