data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha \\in (0, \\frac{\\pi}{2})$ and $\\beta \\in (0, \\frac{\\pi}{2})$, and $\\sin(2\\alpha + \\beta) = \\frac{3}{2} \\sin(\\beta)$, find the minimum value of $\\cos(\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"index": 14600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha \\in \\left( 0, \\pi \\right)$ and $3\\cos 2\\alpha = \\sin \\left( \\frac{\\pi}{4} - \\alpha \\right)$, find the value of $\\sin 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}{18}"
} | {
"index": 14601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha \\in \\left(\\frac{\\pi}{2}, \\pi\\right)$ and $\\sin \\alpha = \\frac{4}{5}$, calculate the value of $\\sin 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{25}"
} | {
"index": 14602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, $\\cos\\alpha=\\frac{{\\sqrt{5}}}{5}$, $\\cos({\\alpha-\\beta})=\\frac{3\\sqrt{10}}{10}$, find the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{10}"
} | {
"index": 14603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, $\\tan\\alpha= \\frac {1}{7}$, $\\sin\\beta= \\frac { \\sqrt {10}}{10}$, find $\\alpha+2\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{4}"
} | {
"index": 14604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, and $\\cos(\\alpha + \\beta) = \\frac{3}{5}$, $\\sin\\alpha = \\frac{5}{13}$, find the value of $\\cos\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 14605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, and $\\cos(\\alpha+\\beta)=\\frac{\\sin\\alpha}{\\sin\\beta}$, determine the maximum value of $\\tan \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 14606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are the roots of $x^2 - 2x - 1 = 0,$ find $5 \\alpha^4 + 12 \\beta^3.$"
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 14607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are the roots of $x^2 - 3x + 1 = 0,$ find $7 \\alpha^5 + 8 \\beta^4.$"
}
] | math | {
"style": "rule",
"ground_truth": "1448"
} | {
"index": 14608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are the roots of the equation $x^2 - 3x - 2 = 0,$ find the value of $5 \\alpha^4 + 12 \\beta^3.$"
}
] | math | {
"style": "rule",
"ground_truth": "672.5 + 31.5\\sqrt{17}"
} | {
"index": 14609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are two interior angles of an oblique triangle, if $\\frac{{\\cos \\alpha - \\sin \\alpha}}{{\\cos \\alpha + \\sin \\alpha}} = \\cos 2\\beta$, then the minimum value of $\\tan \\alpha + \\tan \\beta$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 14610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an acute angle and satisfies $\\cos(\\alpha+\\frac{\\pi}{4})=\\frac{\\sqrt{3}}{3}$. \n$(1)$ Find the value of $\\sin(\\alpha+\\frac{7\\pi}{12})$. \n$(2)$ Find the value of $\\cos(2\\alpha+\\frac{\\pi}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}-1}{6}"
} | {
"index": 14611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an acute angle, $\\beta$ is an obtuse angle, $\\cos\\alpha=\\frac{3}{5}$, $\\sin\\beta=\\frac{5}{13}$, \n$(1)$ Find $\\sin(\\alpha - \\beta)$ and $\\cos(\\alpha - \\beta)$; \n$(2)$ Find the value of $\\tan 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{7}"
} | {
"index": 14612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, and $P(x, 4)$ is a point on its terminal side, with $\\cos\\alpha= \\frac {1}{5}x$, then $x= \\_\\_\\_\\_\\_\\_$, and $\\tan\\alpha= \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac {4}{3}"
} | {
"index": 14613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, and $\\cos\\left( \\frac {\\pi}{2}-\\alpha\\right)= \\frac {4}{5}$, then $\\tan\\alpha= \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac {4}{3}"
} | {
"index": 14614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, simplify $$\\frac { \\sqrt {1+2\\sin(5\\pi-\\alpha)\\cos(\\alpha-\\pi)}}{\\sin\\left(\\alpha - \\frac {3}{2}\\pi \\right)- \\sqrt {1-\\sin^{2}\\left( \\frac {3}{2}\\pi+\\alpha\\right)}}.$$"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, simplify the function $f(\\alpha) = \\frac{\\tan(\\alpha - \\pi) \\cos(2\\pi - \\alpha) \\sin(-\\alpha + \\frac{3\\pi}{2})}{\\cos(-\\alpha - \\pi) \\tan(\\pi + \\alpha)}$. Then, if $\\cos(\\alpha + \\frac{\\pi}{2}) = -\\f... | math | {
"style": "rule",
"ground_truth": "-\\frac{2\\sqrt{6}}{5}"
} | {
"index": 14616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, $f(\\alpha) = \\frac {\\sin(\\pi-\\alpha)\\cdot \\cos(2\\pi-\\alpha)\\cdot \\tan(-\\alpha-\\pi)}{\\tan(-\\alpha )\\cdot \\sin(-\\pi -\\alpha)}$.\n1. Simplify $f(\\alpha)$;\n2. If $\\cos\\left(\\alpha- \\frac {3}{2}\\pi\\ri... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 14617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, $f\\left( \\alpha \\right)=\\dfrac{\\sin (\\alpha -\\dfrac{\\pi }{2})\\cos (\\dfrac{3\\pi }{2}+\\alpha )\\tan (\\pi -\\alpha )}{\\tan (-\\alpha -\\pi )\\sin (-\\alpha -\\pi )}$.\n\n(1) Simplify $f\\left( \\alpha \\right)$\... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{2\\sqrt{6}}{5}"
} | {
"index": 14618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, and $f(\\alpha)=\\frac{{\\sin({\\frac{\\pi}{2}-\\alpha})\\cos(-\\alpha)\\tan(\\pi+\\alpha)}}{{\\cos(\\pi-\\alpha)}}$. \n$(1)$ Simplify $f(\\alpha)$; \n$(2)$ If $f(\\alpha)=\\frac{{2\\sqrt{5}}}{5}$, find the value of $\\c... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{5}"
} | {
"index": 14619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, the function $f(\\alpha)$ is defined as:\n$$f(\\alpha) = \\frac {\\sin(\\alpha - \\frac {\\pi}{2}) \\cdot \\cos( \\frac {3\\pi}{2} + \\alpha) \\cdot \\tan(\\pi - \\alpha)}{\\tan(-\\alpha - \\pi) \\cdot \\sin(-\\alpha - \\p... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2} - 1}{10}"
} | {
"index": 14620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha$ is an obtuse angle and $\\beta$ is also an obtuse angle with $\\cos\\alpha = -\\frac{2\\sqrt{5}}{5}$ and $\\sin\\beta = \\frac{\\sqrt{10}}{10}$, find the value of $\\alpha + \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\pi}{4}"
} | {
"index": 14621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha, \\beta, \\gamma$ are all acute angles and $\\cos^{2} \\alpha + \\cos^{2} \\beta + \\cos^{2} \\gamma = 1$, find the minimum value of $\\tan \\alpha \\cdot \\tan \\beta \\cdot \\tan \\gamma$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha^{2005}+\\beta^{2005}$ can be expressed as a two-variable polynomial in terms of $\\alpha + \\beta$ and $\\alpha \\beta$, find the sum of the coefficients of this polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\angle A$ and $\\angle B$ are within the interval $\\left(0, \\frac{\\pi}{2}\\right)$, and that $\\frac{\\sin A}{\\sin B} = \\sin (A+B)$, find the maximum value of $\\tan A$."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 14624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\angle BAC = 90^{\\circ}$ and the quadrilateral $ADEF$ is a square with side length 1, find the maximum value of $\\frac{1}{AB} + \\frac{1}{BC} + \\frac{1}{CA}$."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{\\sqrt{2}}{2}"
} | {
"index": 14625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\angle C=90^{\\circ}$, \\(A D=D B\\), \\(D E \\perp A B\\), \\(A B=20\\), and \\(A C=12\\), find the area of quadrilateral \\(AD EC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "58 \\frac{1}{2}"
} | {
"index": 14626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{17}{10}=19448$, $\\binom{17}{11}=12376$ and $\\binom{19}{12}=50388$, find $\\binom{18}{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "18564"
} | {
"index": 14627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{17}{7}=19448$, $\\binom{17}{8}=24310$ and $\\binom{17}{9}=24310$, calculate $\\binom{19}{9}$."
}
] | math | {
"style": "rule",
"ground_truth": "92378"
} | {
"index": 14628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{18}{7}=31824$, $\\binom{18}{8}=43758$ and $\\binom{18}{9}=43758$, calculate $\\binom{20}{9}$."
}
] | math | {
"style": "rule",
"ground_truth": "163098"
} | {
"index": 14629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{18}{8}=31824$, $\\binom{18}{9}=48620$, and $\\binom{18}{10}=43758$, calculate $\\binom{20}{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "172822"
} | {
"index": 14630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{20}{13} = 77520$, $\\binom{20}{14} = 38760$ and $\\binom{18}{12} = 18564$, find $\\binom{19}{13}$."
}
] | math | {
"style": "rule",
"ground_truth": "27132"
} | {
"index": 14631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{21}{13}=20349$, $\\binom{21}{14}=11628$, and $\\binom{23}{15}=490314$, find $\\binom{22}{15}$."
}
] | math | {
"style": "rule",
"ground_truth": "458337"
} | {
"index": 14632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{23}{3}=1771$, $\\binom{23}{4}=8855$, and $\\binom{23}{5}=33649$, find $\\binom{25}{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "53130"
} | {
"index": 14633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{23}{5}=33649$, $\\binom{23}{6}=42504$, and $\\binom{23}{7}=33649$, find $\\binom{25}{7}$."
}
] | math | {
"style": "rule",
"ground_truth": "152306"
} | {
"index": 14634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{24}{3}=2024$, $\\binom{24}{4}=10626$, and $\\binom{24}{5}=42504$, find $\\binom{26}{6}$."
}
] | math | {
"style": "rule",
"ground_truth": "230230"
} | {
"index": 14635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\binom{24}{5}=42504$, and $\\binom{24}{6}=134596$, find $\\binom{26}{6}$."
}
] | math | {
"style": "rule",
"ground_truth": "230230"
} | {
"index": 14636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos ( \\frac {π}{6}+α) \\cdot \\cos ( \\frac {π}{3}-α)=- \\frac {1}{4}$, where $α \\in ( \\frac {π}{3}, \\frac {π}{2})$, find the value of $\\sin 2α$ and the value of $\\tan α - \\frac {1}{\\tan α}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 14637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos (\\alpha - \\frac{\\pi }{3}) - \\cos \\alpha = \\frac{1}{3}$, find the value of $\\sin (\\alpha - \\frac{\\pi }{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos (\\frac{\\pi}{4}-a) = \\frac{12}{13}$ and $(\\frac{\\pi}{4}-a)$ is a first-quadrant angle, find the value of $\\frac{\\sin (\\frac{\\pi}{2}-2a)}{\\sin (\\frac{\\pi}{4}+a)}$: \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{119}{144}"
} | {
"index": 14639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos (α+ \\dfrac {π}{6})= \\dfrac {1}{3}$, where $α∈(0\\,,\\,\\, \\dfrac {π}{2})$, find $\\sin α$ and $\\sin (2α+ \\dfrac {5π}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac {7}{9}"
} | {
"index": 14640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos (α- \\frac {π}{6})+ \\sin α= \\frac {4}{5} \\sqrt {3}$, and $α \\in (0, \\frac {π}{3})$, find the value of $\\sin (α+ \\frac {5}{12}π)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7 \\sqrt{2}}{10}"
} | {
"index": 14641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos (α-β)= \\frac{3}{5}$, $\\sin β=- \\frac{5}{13}$, and $α∈\\left( \\left. 0, \\frac{π}{2} \\right. \\right)$, $β∈\\left( \\left. - \\frac{π}{2},0 \\right. \\right)$, find the value of $\\sin α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"index": 14642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos 78^{\\circ}$ is approximately $\\frac{1}{5}$, $\\sin 66^{\\circ}$ is approximately:"
}
] | math | {
"style": "rule",
"ground_truth": "0.92"
} | {
"index": 14643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha + \\sin \\alpha = \\frac{2}{3}$, find the value of $\\frac{\\sqrt{2}\\sin(2\\alpha - \\frac{\\pi}{4}) + 1}{1 + \\tan \\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"index": 14644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha = -\\frac{4}{5}$, and $\\alpha$ is an angle in the third quadrant, find the values of $\\sin \\alpha$ and $\\tan \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha = \\dfrac{1}{7}$, $\\cos (\\alpha - \\beta) = \\dfrac{13}{14}$, and $0 < \\beta < \\alpha < \\dfrac{\\pi}{2}$.\n\n1. Find the value of $\\tan 2\\alpha$.\n2. Find the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"index": 14646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha =-\\dfrac{3}{4}, \\sin \\beta =\\dfrac{2}{3}$, with $\\alpha$ in the third quadrant and $\\beta \\in (\\dfrac{\\pi }{2}, \\pi )$.\n\n(I) Find the value of $\\sin 2\\alpha$;\n\n(II) Find the value of $\\cos (2\\alpha + \\beta )$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{5} + 6\\sqrt{7}}{24}"
} | {
"index": 14647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha =\\dfrac{4}{5}$ and $\\cos (\\alpha +\\beta )=\\dfrac{5}{13}$, where $\\alpha$ and $\\beta$ are acute angles.\n1. Find the value of $\\sin 2\\alpha$.\n2. Find the value of $\\sin \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{33}{65}"
} | {
"index": 14648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha =\\dfrac{\\sqrt{5}}{5}$ and $\\sin (\\alpha -\\beta )=\\dfrac{\\sqrt{10}}{10}$, calculate the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"index": 14649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\alpha$ is a root of the equation $3x^2 - x - 2 = 0$ and $\\alpha$ is an angle in the third quadrant, find the value of $\\frac{\\sin (-\\alpha + \\frac{3\\pi}{2}) \\cos (\\frac{3\\pi}{2} + \\alpha) \\tan^2 (\\pi - \\alpha)}{\\cos (\\frac{\\pi}{2} + \\alpha) ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 14650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\left(\\alpha+ \\frac{\\pi}{6}\\right)= \\frac{4}{5}$, find the value of $\\sin \\left(2\\alpha+ \\frac{\\pi}{3}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{25}"
} | {
"index": 14651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos \\theta = \\frac{12}{13}, \\theta \\in \\left( \\pi, 2\\pi \\right)$, find the values of $\\sin \\left( \\theta - \\frac{\\pi}{6} \\right)$ and $\\tan \\left( \\theta + \\frac{\\pi}{4} \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{17}"
} | {
"index": 14652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos x - 4 \\sin x = 1,$ find all possible values of $\\sin x + 4 \\cos x.$"
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 14653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos α= \\frac{1}{7}, \\cos (α+β)=- \\frac{11}{14}$, and $α,β∈(0, \\frac{π}{2})$, find the value of $\\cos β$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos α=-\\dfrac{4}{5}\\left(\\dfrac{π}{2}<α<π\\right)$, find $\\cos\\left(\\dfrac{π}{6}-α\\right)$ and $\\cos\\left(\\dfrac{π}{6}+α\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{3+4\\sqrt{3}}{10}"
} | {
"index": 14655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(3\\pi + \\alpha) = \\frac{3}{5}$, find the values of $\\cos(\\alpha)$, $\\cos(\\pi + \\alpha)$, and $\\sin(\\frac{3\\pi}{2} - \\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(75^\\circ + \\alpha) = \\frac{1}{3}$, where $\\alpha$ is an angle in the third quadrant, find the value of $\\cos(105^\\circ - \\alpha) + \\sin(\\alpha - 105^\\circ)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} - 1}{3}"
} | {
"index": 14657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(\\alpha - \\beta) = \\frac{3}{5}$, $\\sin(\\beta) = -\\frac{5}{13}$, where $\\alpha \\in \\left(0, \\frac{\\pi}{2} \\right)$, $\\beta \\in \\left(-\\frac{\\pi}{2}, 0 \\right)$, find the value of $\\sin(\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"index": 14658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(\\alpha+\\pi)= \\frac {3}{5}$ and $\\pi \\leq \\alpha < 2\\pi$, calculate the value of $\\sin(-\\alpha-2\\pi)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 14659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(\\frac{\\pi}{6} - \\alpha) = \\frac{1}{3}$, determine the value of $\\sin(\\frac{5\\pi}{6} - 2\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"index": 14660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(\\frac{\\pi}{6} - \\alpha) = \\frac{3}{5}$, find the value of $\\cos(\\frac{5\\pi}{6} + \\alpha)$:\n\nA) $\\frac{3}{5}$\nB) $-\\frac{3}{5}$\nC) $\\frac{4}{5}$\nD) $-\\frac{4}{5}$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"index": 14661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(\\frac{\\pi}{6} - \\alpha) = \\frac{\\sqrt{3}}{3}$, find the value of $\\sin(\\frac{5\\pi}{6} - 2\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 14662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(\\pi+\\theta) = -\\frac{1}{2}$, find the value of $\\tan(\\theta - 9\\pi)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos(x - \\frac{\\pi}{4}) = \\frac{\\sqrt{2}}{10}$, with $x \\in (\\frac{\\pi}{2}, \\frac{3\\pi}{4})$.\n(1) Find the value of $\\sin x$;\n(2) Find the value of $\\cos(2x - \\frac{\\pi}{3})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7 + 24\\sqrt{3}}{50}"
} | {
"index": 14664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos\\alpha = \\frac{1}{7}$, $\\cos(\\alpha-\\beta) = \\frac{13}{14}$, and $0<\\beta<\\alpha<\\frac{\\pi}{2}$,\n(1) find the value of $\\tan 2\\alpha$;\n(2) determine $\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos\\left(\\frac {\\pi}{4} + \\theta\\right) = -\\frac {3}{5}$, and $\\frac {11\\pi}{12} < \\theta < \\frac {5\\pi}{4}$, find the value of $\\frac {\\sin{2\\theta} + 2\\sin^{2}{\\theta}}{1 - \\tan{\\theta}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {28}{75}"
} | {
"index": 14666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\cos{\\alpha}=-\\frac{4}{5}$, where $\\alpha$ is an angle in the third quadrant, find the value of $\\sin{\\left(\\alpha-\\frac{\\pi}{4}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{10}"
} | {
"index": 14667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\dfrac {\\pi}{2} < \\alpha < \\pi$ and $\\sin (\\alpha+ \\dfrac {\\pi}{6})= \\dfrac {3}{5}$, find the value of $\\cos (\\alpha- \\dfrac {\\pi}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {3\\sqrt {3}-4}{10}"
} | {
"index": 14668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\dfrac {\\pi}{4} < \\alpha < \\dfrac {3\\pi}{4}$ and $0 < \\beta < \\dfrac {\\pi}{4}$, with $\\cos \\left( \\dfrac {\\pi}{4}+\\alpha \\right)=- \\dfrac {3}{5}$ and $\\sin \\left( \\dfrac {3\\pi}{4}+\\beta \\right)= \\dfrac {5}{13}$, find the value of $\\sin(\\alpha... | math | {
"style": "rule",
"ground_truth": "\\dfrac {63}{65}"
} | {
"index": 14669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\dfrac {π}{2} < β < α < \\dfrac {3}{4}π$, $\\cos (α+β)=- \\dfrac {3}{5}$, and $\\sin (α-β)= \\dfrac {5}{13}$, find $\\cos 2β$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\dfrac {56}{65}"
} | {
"index": 14670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\dfrac{\\pi}{2} < \\alpha < \\beta < \\dfrac{3\\pi}{4}, \\cos(\\alpha - \\beta) = \\dfrac{12}{13}, \\sin(\\alpha + \\beta) = -\\dfrac{3}{5}$, find the value of $\\sin 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{56}{65}"
} | {
"index": 14671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac 1n - \\frac{1}{n+1} < \\frac{1}{10}$, what is the least possible positive integer value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 14672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac {π}{2}＜α＜ \\frac {3π}{2}$, points A, B, and C are in the same plane rectangular coordinate system with coordinates A(3, 0), B(0, 3), and C(cosα, sinα) respectively.\n(1) If $| \\overrightarrow {AC}|=| \\overrightarrow {BC}|$, find the value of angle α;\n(2) W... | math | {
"style": "rule",
"ground_truth": "- \\frac {5}{9}"
} | {
"index": 14673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{3\\pi}{4} < \\alpha < \\pi$, find the values for the following:\n1. $\\tan\\alpha$ given $\\tan\\alpha + \\frac{1}{\\tan\\alpha} = -\\frac{10}{3}$.\n2. The value of $$\\frac{5\\sin^2\\left(\\frac{\\alpha}{2}\\right) + 8\\sin\\left(\\frac{\\alpha}{2}\\right)\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{4}"
} | {
"index": 14674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{5+7+9}{3} = \\frac{4020+4021+4022}{M}$, find $M$."
}
] | math | {
"style": "rule",
"ground_truth": "1723"
} | {
"index": 14675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{\\cos 2\\alpha}{\\sqrt{2}\\sin\\left(\\alpha+\\frac{\\pi}{4}\\right)}=\\frac{\\sqrt{5}}{2}$, find the value of $\\tan\\alpha+\\frac{1}{\\tan\\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 14676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{\\cos \\alpha + \\sin \\alpha}{\\cos \\alpha - \\sin \\alpha} = 2$, find the value of $\\frac{1 + \\sin 4\\alpha - \\cos 4\\alpha}{1 + \\sin 4\\alpha + \\cos 4\\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{\\cos \\alpha}{1 + \\sin \\alpha} = \\sqrt{3}$, find the value of $\\frac{\\cos \\alpha}{\\sin \\alpha - 1}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{3}"
} | {
"index": 14678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{\\pi}{4} < \\alpha < \\frac{3\\pi}{4}$, $0 < \\beta < \\frac{\\pi}{4}$, $\\sin(\\alpha + \\frac{\\pi}{4}) = \\frac{3}{5}$, and $\\cos(\\frac{\\pi}{4} + \\beta) = \\frac{5}{13}$, find the value of $\\sin(\\alpha + \\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 14679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{\\sin \\theta + \\cos \\theta}{\\sin \\theta - \\cos \\theta} = 2$, find the value of $\\frac{\\sin \\theta}{\\cos^{3} \\theta} + \\frac{\\cos \\theta}{\\sin^{3} \\theta}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{820}{27}"
} | {
"index": 14680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{a}{25-a}+\\frac{b}{65-b}+\\frac{c}{60-c}=7$, evaluate $\\frac{5}{25-a}+\\frac{13}{65-b}+\\frac{12}{60-c}$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 14681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{a}{36-a}+\\frac{b}{48-b}+\\frac{c}{72-c}=9$, evaluate $\\frac{4}{36-a}+\\frac{6}{48-b}+\\frac{9}{72-c}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{3}"
} | {
"index": 14682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{x}{2} = y^2$ and $\\frac{x}{5} = 3y$, solve for $x$."
}
] | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"index": 14683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{x}{3} = y^2$ and $\\frac{x}{6} = 3y$, solve for the value of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 14684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{x}{9}, \\frac{y}{15}, \\frac{z}{14}$ are all in their simplest forms and their product is $\\frac{1}{6}$, find the value of $x+y+z$."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 14685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\frac{{\\cos 2\\alpha}}{{\\sin(\\alpha+\\frac{\\pi}{4})}}=\\frac{4}{7}$, find the value of $\\sin 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{49}"
} | {
"index": 14686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\left\\{a_{n}\\right\\}$ is an arithmetic sequence with a nonzero common difference and $\\left\\{b_{n}\\right\\}$ is a geometric sequence where $a_{1}=3, b_{1}=1, a_{2}=b_{2}, 3a_{5}=b_{3}$, and there exist constants $\\alpha$ and $\\beta$ such that for every posi... | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{3} + 3"
} | {
"index": 14687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\lg 2 = 0.3010$, determine the number of digits in the integer $2^{2015}$."
}
] | math | {
"style": "rule",
"ground_truth": "607"
} | {
"index": 14688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\log_{10} \\sin x + \\log_{10} \\cos x = -1$ and that $\\log_{10} (\\sin x + \\cos x) = \\frac{1}{2} (\\log_{10} n - 1),$ find $n.$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14689,
"solution": "Using the properties of logarithms, we can simplify the first equation to $\\log_{10} \\sin x + \\log_{10} \\cos x = \\log_{10}(\\sin x \\cos x) = -1$. Therefore, \\[\\sin x \\cos x = \\frac{1}{10}.\\qquad (*)\\]\n Now, manipulate the second equation. \\begin{align*} \\log_{10} (\\sin... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\log_{10}\\sin x + \\log_{10}\\cos x= -1$ and that $\\log_{10}(\\sin x+\\cos\nx)=\\frac{1}{2}(\\log_{10}n-1)$, find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\log_{10}\\sin x + \\log_{10}\\cos x= -2$ and that $\\log_{10}(\\sin x+\\cos x)=\\frac{1}{2}(\\log_{10}m-2)$, find $m$."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 14691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\mathbf{a}$ and $\\mathbf{b}$ are nonzero vectors such that $\\|\\mathbf{a} + \\mathbf{b}\\| = \\|\\mathbf{a} - \\mathbf{b}\\|,$ find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "90^\\circ"
} | {
"index": 14692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\operatorname{log}_{8}(p) = \\operatorname{log}_{12}(q) = \\operatorname{log}_{18}(p-q)$, calculate the value of $\\frac{q}{p}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 14693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\operatorname{tg} \\theta$ and $\\operatorname{ctg} \\theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\\alpha < \\theta < \\frac{5 \\pi}{4}$, find the value of $\\cos \\theta - \\sin \\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{\\frac{5 - 2\\sqrt{5}}{5}}"
} | {
"index": 14694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overline{2 a 1 b 9}$ represents a five-digit number, how many ordered digit pairs $(a, b)$ are there such that\n$$\n\\overline{2 a 1 b 9}^{2019} \\equiv 1 \\pmod{13}?\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 14695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow {a}|=4$, $\\overrightarrow {e}$ is a unit vector, and the angle between $\\overrightarrow {a}$ and $\\overrightarrow {e}$ is $\\frac {2π}{3}$, find the projection of $\\overrightarrow {a}+ \\overrightarrow {e}$ on $\\overrightarrow {a}- \\overrighta... | math | {
"style": "rule",
"ground_truth": "\\frac {5 \\sqrt {21}}{7}"
} | {
"index": 14696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow {e_{1}}$ and $\\overrightarrow {e_{2}}$ are unit vectors with an angle of $\\frac {2π}{3}$ between them, and $\\overrightarrow {a}$ = 3 $\\overrightarrow {e_{1}}$ + 2 $\\overrightarrow {e_{2}}$, $\\overrightarrow {b}$ = 3 $\\overrightarrow {e_{2}}$,... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 14697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{AB} \\perp \\overrightarrow{AC}$, $|\\overrightarrow{AB}|= \\frac{1}{t}$, $|\\overrightarrow{AC}|=t$, and point $P$ is a point on the plane of $\\triangle ABC$ such that $\\overrightarrow{AP}= \\frac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|} + \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{OA}$, $\\overrightarrow{OB}$, and $\\overrightarrow{OC}$ are all unit vectors, and satisfy $\\frac{1}{2}\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$, calculate the value of $\\overrightarrow{AB} \\cdot \\overri... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 14699,
"solution": ""
} |
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