{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"b20d4713-70df-5929-9868-42a288abe09b","task_key":"precalculus--test--b20d4713-70df-5929-9868-42a288abe09b","task_revision_id":"1","upstream_id":"","short_description":"For $\\pi\\leq\\theta<2\\pi$, let","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"For $\\\\pi\\\\leq\\\\theta<2\\\\pi$, let\\n\\\\[ P=\\\\dfrac12\\\\cos\\\\theta-\\\\dfrac14\\\\sin2\\\\theta-\\\\dfrac18\\\\cos3\\\\theta+\\\\dfrac1{16}\\\\sin4\\\\theta+\\\\dfrac1{32}\\\\cos5\\\\theta-\\\\dfrac1{64}\\\\sin6\\\\theta-\\\\dfrac1{128}\\\\cos7\\\\theta+\\\\dotsb\\n\\\\]and  \\\\[ Q=1-\\\\dfrac12\\\\sin\\\\theta-\\\\dfrac14\\\\cos2\\\\theta+\\\\dfrac1{8}\\\\sin3\\\\theta+\\\\dfrac1{16}\\\\cos4\\\\theta-\\\\dfrac1{32}\\\\sin5\\\\theta-\\\\dfrac1{64}\\\\cos6\\\\theta+\\\\dfrac1{128}\\\\sin7\\\\theta\\n+\\\\dotsb\\\\]so that $\\\\frac PQ = \\\\frac{2\\\\sqrt2}7$.  Find $\\\\sin\\\\theta.$\",\"type\":\"Precalculus\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}