The solutions for \(\theta\) in the interval \([0^\circ, 360^\circ]\) are:

["Solutions for (\ heta) in the Interval ([0^\circ, 360^\circ]): A Complete Guide to Solving Trigonometric Equations", "When solving trigonometric equations involving the angle (\ heta) measured in degrees within the interval ([0^\circ, 360^\circ]), understanding and applying the correct solution techniques is essential. Whether you're studying sine, cosine, or tangent functions, knowing how to find all valid angular solutions in this full circle range helps ensure accuracy and deepens your grasp of periodic functions.", "In this comprehensive article, we explore the solutions for (\ heta) in ([0^\circ, 360^\circ]), explain how to identify and compute them across all major trigonometric functions, and highlight common approaches and pitfalls to avoid.", "---", "### Why Focus on ([0^\circ, 360^\circ])?", "The interval ([0^\circ, 360^\circ]) represents one full rotation of the unit circle, allowing all unique angle measures from start to finish. This range is standard in trigonometry for solving equations because it captures all periodic behaviors without redundancy and provides context for interpreting sine, cosine, and tangent functions as repeating patterns.", "---", "### General Steps to Solve (\ heta) in ([0^\circ, 360^\circ])", "1. Rewrite the Equation: Express trigonometric relationships based on known values or identities.\n2. Find Reference Angles: Solve the equation using the reference angle in the first quadrant.\n3. Determine All Quadrants: Use symmetry and periodicity to find solutions in all four quadrants.\n4. Adjust for the Given Interval: Ensure solutions lie within ([0^\circ, 360^\circ]).", "---", "### Solving by Trigonometric Function", "#### 1. Solve (\sin \ heta = k)", "- Use the inverse sine function:\n [\n \ heta = \arcsin(k)\n ]\n- For (\sin \ heta = k), one solution is:\n [\n \ heta = \arcsin(k)\n ]\n- Since sine is positive in quadrants I and II:\n [\n \ heta = \arcsin(k) \quad \ ext{and} \quad \ heta = 180^\circ - \arcsin(k)\n ]\n- All solutions are in ([0^\circ, 360^\circ]), provided (\arcsin(k)) is defined ((-1 \leq k \leq 1)).", "Example: Solve (\sin \ heta = \frac{1}{2})\n- (\arcsin\left(\frac{1}{2}\right) = 30^\circ)\n- Solutions: (\ heta = 30^\circ, 150^\circ)", "---", "#### 2. Solve (\cos \ heta = k)", "- One solution:\n [\n \ heta = \arccos(k)\n ]\n- Cosine is positive in quadrants I and IV; negative in II and III.\n- Solutions:\n [\n \ heta = \arccos(k), \quad 360^\circ - \arccos(k)\n ]", "Example: Solve (\cos \ heta = -\frac{\sqrt{2}}{2})\n- (\arccos\left(-\frac{\sqrt{2}}{2}\right) = 135^\circ)\n- Solutions: (\ heta = 135^\circ, 225^\circ)", "---", "#### 3. Solve (\ an \ heta = k)", "- One principal solution:\n [\n \ heta = \arctan(k)\n ]\n- Tangent is positive in quadrants I and III; negative in II and IV.\n- General solutions (accounting for (180^\circ) periodicity of tangent):\n [\n \ heta = \arctan(k) + 180^\circ \cdot n, \quad n \in \mathbb{Z}\n ]\n- Within ([0^\circ, 360^\circ]):\n [\n \ heta = \arctan(k) \quad \ ext{and} \quad \ heta = \arctan(k) + 180^\circ\n ]", "Example: Solve (\ an \ heta = 1)\n- (\arctan(1) = 45^\circ)\n- Solutions: (45^\circ) and (225^\circ)", "---", "#### 4. Solve (\cot \ heta = k)", "- Cotangent is the reciprocal of tangent, so:\n [\n \ heta = \arccot(k) = \arctan\left(\frac{1}{k}\right), \quad n \in \mathbb{Z}\n ]\n- Equivalent to:\n [\n \ heta = 180^\circ - \arctan(k), \quad \ ext{and} \quad \ heta = 360^\circ + \arctan\left(\frac{1}{k}\right)\n ]\n- Solutions within ([0^\circ, 360^\circ]):\n [\n \ heta = \arccot(k) \quad \ ext{and} \quad 180^\circ - \arccot(k)\n ]", "---", "### Tips & Common Mistakes", "- Use Reference Angles: Always compute the reference angle first; then apply signs based on quadrant.\n- Check Domain: Ensure solutions fall within ([0^\circ, 360^\circ]); periodic functions repeat, but only want unique principal solutions.\n- Trig Ratios Under Special Values: Memorize exact values for (0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ) to speed up solutions.\n- Avoid Common Errors: Confusion between (\sin) and (\cos) signs, or forgetting tangent’s (180^\circ) periodicity.", "---", "### Summary Table: Key Solutions in ([0^\circ, 360^\circ])", "| Function | Solution Steps | General Solutions in ([0°, 360°]) |\n|----------|------------------------------------------|----------------------------------------------------|\n| (\sin \ heta = k) | (\ heta = \arcsin(k)) + (180^\circ) offset | (\arcsin(k)), (180^\circ - \arcsin(k)) |\n| (\cos \ heta = k) | (\ heta = \arccos(k)), (360^\circ - \arccos(k)) | (\arccos(k)), (360^\circ - \arccos(k)) |\n| (\ an \ heta = k) | (\ heta = \arctan(k) + 180^\circ \ imes n) | (\arctan(k) \mod 360^\circ), (\arctan(k) + 180^\circ) |\n| (\cot \ heta = k) | Equivalent to (\ an(90^\circ - \ heta)) or use reciprocal | (180^\circ - \arctan(k)), (360^\circ - \arctan(k)) |", "---", "### Conclusion", "Mastering the solutions for (\ heta) in the interval ([0^\circ, 360^\circ]) is fundamental in trigonometry. By combining inverse function evaluations with symmetry and periodic properties, students can confidently solve any trigonometric equation in this fundamental range. Whether for homework, exams, or applied fields like engineering or physics, understanding these solutions ensures both accuracy and clarity when working with angular measurements.", "Start practicing with varied values of (k) to build fluency—remember, the unit circle is your best ally in mastering (\ heta) solutions!", "---", "Keywords: solutions for theta in degrees, solve trigonometric equations, (\sin \ heta = k), (\cos \ heta = k), (\ an \ heta = k), solutions in [0°, 360°], unit circle, trigonometric identities, inverse trig functions."]