+ data = {"data_row": {"id": "clwb6wvpv3mpx0712aafl9m00", "external_id": "43cdad5e-1fcf-450d-ad72-df4460edf973", "global_key": "9ab56c5a-5c2f-45ae-8e21-e53eb415cefe", "row_data": "{\"type\":\"application/vnd.labelbox.conversational\",\"version\":1,\"messages\":[{\"messageId\":\"message-0\",\"timestampUsec\":1530718491,\"content\":\"The minimum value of $3 \\\\cos x + 4 \\\\sin x + 8$ is\",\"user\":{\"userId\":\"prompt\",\"name\":\"prompt\"},\"align\":\"left\",\"canLabel\":true}],\"modelOutputs\":[{\"title\":\"Response 1\",\"content\":\"To find the minimum value of the expression $3 \\\\cos x + 4 \\\\sin x + 8$, we can use the fact that $a\\\\cos x+b\\\\sin x=\\\\sqrt{a^2+b^2}\\\\left(\\\\frac{a}{\\\\sqrt{a^2+b^2}}\\\\cos x+\\\\frac{b}{\\\\sqrt{a^2+b^2}}\\\\sin x\\\\right)$. This allows us to rewrite the expression as:\\n\\n$3\\\\cos x+4\\\\sin x+8=\\\\sqrt{3^2+4^2}\\\\left(\\\\frac{3}{\\\\sqrt{3^2+4^2}}\\\\cos x+\\\\frac{4}{\\\\sqrt{3^2+4^2}}\\\\sin x\\\\right)+8=5\\\\left(\\\\frac{3}{5}\\\\cos x+\\\\frac{4}{5}\\\\sin x\\\\right)+8$\\n\\nNow, let's consider the expression $\\\\frac{3}{5}\\\\cos x+\\\\frac{4}{5}\\\\sin x$. Since $\\\\left(\\\\frac{3}{5}\\\\right)^2+\\\\left(\\\\frac{4}{5}\\\\right)^2=1$, we can write $\\\\frac{3}{5}=\\\\cos\\\\theta$ and $\\\\frac{4}{5}=\\\\sin\\\\theta$ for some angle $\\\\theta$. Then:\\n\\n$\\\\frac{3}{5}\\\\cos x+\\\\frac{4}{5}\\\\sin x=\\\\cos\\\\theta\\\\cos x+\\\\sin\\\\theta\\\\sin x=\\\\cos(x-\\\\theta)$\\n\\nSo, the original expression can be written as:\\n\\n$5\\\\cos(x-\\\\theta)+8$\\n\\nSince the minimum value of $\\\\cos(x-\\\\theta)$ is $-1$, the minimum value of the original expression is:\\n\\n$5(-1)+8=-5+8=3$\\n\\nTherefore, the minimum value of $3\\\\cos x + 4\\\\sin x + 8$ is $\\\\boxed{3}$.\",\"modelConfigName\":\"null\"},{\"title\":\"Response 2\",\"content\":\"A nice math question!\\n\\nTo find the minimum value of $3 \\\\cos x + 4 \\\\sin x + 8$, we can use the fact that $a\\\\cos x + b\\\\sin x = \\\\sqrt{a^2 + b^2} \\\\cos(x - \\\\alpha)$, where $\\\\alpha = \\\\tan^{-1}\\\\left(\\\\frac{b}{a}\\\\right)$.\\n\\nIn this case, $a = 3$ and $b = 4$, so $\\\\alpha = \\\\tan^{-1}\\\\left(\\\\frac{4}{3}\\\\right)$.\\n\\nSo, we have:\\n\\n$$3 \\\\cos x + 4 \\\\sin x + 8 = \\\\sqrt{3^2 + 4^2} \\\\cos(x - \\\\alpha) + 8 = 5 \\\\cos(x - \\\\alpha) + 8$$\\n\\nNow, the minimum value of $\\\\cos(x - \\\\alpha)$ is $-1$, so the minimum value of $5 \\\\cos(x - \\\\alpha) + 8$ is:\\n\\n$$5(-1) + 8 = -5 + 8 = 3$$\\n\\nTherefore, the minimum value of $3 \\\\cos x + 4 \\\\sin x + 8$ is $\\\\boxed{3}$.\\n\\nLet me know if you have any questions or need further clarification!\",\"modelConfigName\":\"null\"}]}"}, "media_attributes": {"asset_type": "conversational", "mime_type": "application/vnd.labelbox.conversational", "labelable_ids": ["message-0"], "message_count": 1}}
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