test_example.json

{
  "problem": "Convert the point $(0,3)$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
  "solution": "We have that $r = \\sqrt{0^2 + 3^2} = 3.$  Also, if we draw the line connecting the origin and $(0,3),$ this line makes an angle of $\\frac{\\pi}{2}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-0.5,0)--(3.5,0));\ndraw((0,-0.5)--(0,3.5));\ndraw(arc((0,0),3,0,90),red,Arrow(6));\n\ndot((0,3), red);\nlabel(\"$(0,3)$\", (0,3), W);\ndot((3,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 3, \\frac{\\pi}{2} \\right)}.$",
  "answer": "\\left( 3, \\frac{\\pi}{2} \\right)",
  "subject": "Precalculus",
  "level": 2,
  "unique_id": "test/precalculus/807.json",
  "data_source": "HuggingFaceH4/MATH-500",
  "prompt": [
    {
      "content": "Convert the point $(0,3)$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Let's think step by step and output the final answer within \\boxed{}.",
      "role": "user"
    }
  ],
  "ability": "math",
  "reward_model": {
    "ground_truth": "\\left( 3, \\frac{\\pi}{2} \\right)",
    "style": "rule"
  },
  "extra_info": {
    "answer": "\\left( 3, \\frac{\\pi}{2} \\right)",
    "index": 0,
    "level": 2,
    "question": "Convert the point $(0,3)$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
    "solution": "We have that $r = \\sqrt{0^2 + 3^2} = 3.$  Also, if we draw the line connecting the origin and $(0,3),$ this line makes an angle of $\\frac{\\pi}{2}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-0.5,0)--(3.5,0));\ndraw((0,-0.5)--(0,3.5));\ndraw(arc((0,0),3,0,90),red,Arrow(6));\n\ndot((0,3), red);\nlabel(\"$(0,3)$\", (0,3), W);\ndot((3,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 3, \\frac{\\pi}{2} \\right)}.$",
    "split": "test",
    "subject": "Precalculus",
    "unique_id": "test/precalculus/807.json"
  }
}