You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
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}}
103
+
data= {
104
+
"data_row": {
105
+
"id":
106
+
"clwb6wvpv3mpx0712aafl9m00",
107
+
"external_id":
108
+
"43cdad5e-1fcf-450d-ad72-df4460edf973",
109
+
"global_key":
110
+
"9ab56c5a-5c2f-45ae-8e21-e53eb415cefe",
111
+
"row_data":
112
+
"{\"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\"}]}"
0 commit comments