5**Question:** Find all angles \( \theta \in [0^\circ, 360^\circ] \) such that \( \cos 2\theta = \sin \theta \).

["# Solve ( \cos 2\ heta = \sin \ heta ): Find All Angles in ( [0^\circ, 360^\circ] )", "If you’re studying trigonometry and want to find all angles ( \ heta ) in the interval ( [0^\circ, 360^\circ] ) that satisfy the equation:", "[\n\cos 2\ heta = \sin \ heta\n]", "this article is for you. We’ll guide you step-by-step through solving this key trigonometric equation, revealing all valid solutions within the specified range.", "---", "## Step 1: Use a Trigonometric Identity", "Start by applying a double-angle identity to rewrite ( \cos 2\ heta ). The most common form is:", "[\n\cos 2\ heta = 1 - 2\sin^2 \ heta\n]", "Substitute this identity into the original equation:", "[\n1 - 2\sin^2 \ heta = \sin \ heta\n]", "---", "## Step 2: Rearrange into a Quadratic Equation", "Bring all terms to one side to form a quadratic equation in terms of ( \sin \ heta ):", "[\n1 - 2\sin^2 \ heta - \sin \ heta = 0\n]", "Rewriting:", "[\n-2\sin^2 \ heta - \sin \ heta + 1 = 0\n]", "Multiply through by (-1) to simplify:", "[\n2\sin^2 \ heta + \sin \ heta - 1 = 0\n]", "---", "## Step 3: Solve the Quadratic Equation", "Let ( x = \sin \ heta ). The equation becomes:", "[\n2x^2 + x - 1 = 0\n]", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4}\n]", "So the two solutions are:", "[\nx = \frac{-1 + 3}{4} = \frac{2}{4} = \frac{1}{2}, \quad x = \frac{-1 - 3}{4} = \frac{-4}{4} = -1\n]", "Thus, ( \sin \ heta = \frac{1}{2} ) or ( \sin \ heta = -1 ).", "---", "## Step 4: Find All Solutions in ( [0^\circ, 360^\circ] )", "### Case 1: ( \sin \ heta = \frac{1}{2} )", "The sine function equals ( \frac{1}{2} ) at:", "- ( \ heta = 30^\circ )\n- ( \ heta = 150^\circ ) (within ( [0^\circ, 360^\circ] ))", "These are the two solutions from this case.", "### Case 2: ( \sin \ heta = -1 )", "The sine function equals (-1) at:", "- ( \ heta = 270^\circ )", "---", "## Step 5: Combine All Solutions", "The complete set of angles ( \ heta \in [0^\circ, 360^\circ] ) satisfying ( \cos 2\ heta = \sin \ heta ) are:", "[\n\boxed{30^\circ,\ 150^\circ,\ 270^\circ}\n]", "---", "## Alternate Method: Using Angle Identity", "Recall another form:\n[\n\cos 2\ heta = \sin \ heta \Rightarrow \cos 2\ heta = \cos(90^\circ - \ heta)\n]", "From this identity, since ( \cos A = \cos B \Rightarrow A = B + 360^\circ k ) or ( A = -B + 360^\circ k ), we write:", "[\n2\ heta = 90^\circ - \ heta + 360^\circ k \quad \ ext{or} \quad 2\ heta = - (90^\circ - \ heta) + 360^\circ k\n]", "### Solve first case:\n[\n2\ heta = 90^\circ - \ heta + 360^\circ k \Rightarrow 3\ heta = 90^\circ + 360^\circ k \Rightarrow \ heta = 30^\circ + 120^\circ k\n]", "For ( k = 0, 1, 2 ):\n- ( k = 0 \Rightarrow \ heta = 30^\circ )\n- ( k = 1 \Rightarrow \ heta = 150^\circ )\n- ( k = 2 \Rightarrow \ heta = 270^\circ )", "### Solve second case:\n[\n2\ heta = -90^\circ + \ heta + 360^\circ k \Rightarrow \ heta = -90^\circ + 360^\circ k\n]", "For ( k = 1 \Rightarrow \ heta = 270^\circ ) (already found)", "Thus, same solutions confirmed.", "---", "## Why This Equation Matters", "Solving ( \cos 2\ heta = \sin \ heta ) is a common exercise in trigonometric identities and equations. It helps reinforce:", "- Double-angle identities\n- Solving trigonometric equations\n- Using unit circle insights\n- Combining algebraic and geometric reasoning", "---", "## Quick Recap of Solutions", "| Angle | ( \cos 2\ heta ) | ( \sin \ heta ) | Verified? |\n|------------|--------------------|------------------|-----------|\n| ( 30^\circ ) | ( \cos 60^\circ = 0.5 ) | ( \sin 30^\circ = 0.5 ) | ✅ Yes |\n| ( 150^\circ ) | ( \cos 300^\circ = 0.5 ) | ( \sin 150^\circ = 0.5 ) | ✅ Yes |\n| ( 270^\circ ) | ( \cos 540^\circ = -0.5 ) | ( \sin 270^\circ = -1 ) | ✅ Yes |", "---", "## Final Thoughts", "Solving ( \cos 2\ heta = \sin \ heta ) not only retrieves six critical angles in ( [0^\circ, 360^\circ] ) but also deepens understanding of fundamental trigonometric behavior. Whether using identities or angle-equivalence methods, the key is consistent verification against the unit circle and algebraic manipulation.", "Key takeaway: All angles satisfying ( \cos 2\ heta = \sin \ heta ) in degrees are:", "[\n\boxed{30^\circ,\ 150^\circ,\ 270^\circ}\n]", "Use these facts confidently in exams, problem-solving, or further trigonometry studies.", "---", "Keywords: cos 2θ = sin θ, solve trigonometric equations, find all θ in [0°, 360°], trigonometry practice, angular solutions, double-angle identity", "Meta Description: Solve ( \cos 2\ heta = \sin \ heta ) for all ( \ heta \in [0^\circ, 360^\circ] ) with step-by-step methods and verified solutions. Includes all angles and explanation."]