{
  "index": "1956-B-4",
  "type": "ANA",
  "tag": [
    "ANA",
    "GEO"
  ],
  "difficulty": "",
  "question": "4. Prove that if \\( A, B \\), and \\( C \\) are angles of a triangle measured in radians then \\( A \\cos B+\\sin A \\cos C>0 \\).",
  "solution": "Solution. We distinguish two cases.\nCase 1. \\( 0<B<\\pi / 2 \\). Then \\( \\cos B>0 \\). Since \\( A>0, A>\\sin A \\); hence \\( A \\cos B>\\sin A \\cos B \\). Also \\( C=\\pi-A-B<\\pi-B \\), and \\( 0<C<\\pi \\), so \\( \\cos C>\\cos (\\pi-B)=-\\cos B \\). Therefore \\( \\cos B+ \\) \\( \\cos C>0 \\). Hence\n\\[\nA \\cos B+\\sin A \\cos C>\\sin A(\\cos B+\\cos C)>0\n\\]\nand the desired inequality is established in Case 1.\nCase 2. \\( \\frac{\\pi}{2} \\leq B<\\pi \\). Then \\( \\cos B \\leq 0,0<A<\\frac{\\pi}{2} \\), so \\( A<\\tan A \\), and \\( \\tan A>0 \\). Hence\n\\[\nA \\cos B \\geq \\tan A \\cos B .\n\\]\n\nAlso \\( B=\\pi-A-C \\). So\n\\[\n\\cos B+\\cos A \\cos C=-\\cos (A+C)+\\cos A \\cos C=\\sin A \\sin B>0 .\n\\]\n\nTherefore\n\\[\n\\begin{array}{l}\nA \\cos B+\\sin A \\cos C \\geq \\tan A \\cos B+\\sin A \\cos C \\\\\n\\quad=\\tan A(\\cos B+\\cos A \\cos C)>0 .\n\\end{array}\n\\]\n\nThus the required inequality is proved in Case 2 also.",
  "vars": [
    "A",
    "B",
    "C"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "anglealpha",
        "B": "anglebeta",
        "C": "anglegamma"
      },
      "question": "4. Prove that if \\( anglealpha, anglebeta \\), and \\( anglegamma \\) are angles of a triangle measured in radians then \\( anglealpha \\cos anglebeta+\\sin anglealpha \\cos anglegamma>0 \\).",
      "solution": "Solution. We distinguish two cases.\nCase 1. \\( 0<anglebeta<\\pi / 2 \\). Then \\( \\cos anglebeta>0 \\). Since \\( anglealpha>0, anglealpha>\\sin anglealpha \\); hence \\( anglealpha \\cos anglebeta>\\sin anglealpha \\cos anglebeta \\). Also \\( anglegamma=\\pi-anglealpha-anglebeta<\\pi-anglebeta \\), and \\( 0<anglegamma<\\pi \\), so \\( \\cos anglegamma>\\cos (\\pi-anglebeta)=-\\cos anglebeta \\). Therefore \\( \\cos anglebeta+ \\) \\( \\cos anglegamma>0 \\). Hence\n\\[\nanglealpha \\cos anglebeta+\\sin anglealpha \\cos anglegamma>\\sin anglealpha(\\cos anglebeta+\\cos anglegamma)>0\n\\]\nand the desired inequality is established in Case 1.\nCase 2. \\( \\frac{\\pi}{2} \\leq anglebeta<\\pi \\). Then \\( \\cos anglebeta \\leq 0,0<anglealpha<\\frac{\\pi}{2} \\), so \\( anglealpha<\\tan anglealpha \\), and \\( \\tan anglealpha>0 \\). Hence\n\\[\nanglealpha \\cos anglebeta \\geq \\tan anglealpha \\cos anglebeta .\n\\]\n\nAlso \\( anglebeta=\\pi-anglealpha-anglegamma \\). So\n\\[\n\\cos anglebeta+\\cos anglealpha \\cos anglegamma=-\\cos (anglealpha+anglegamma)+\\cos anglealpha \\cos anglegamma=\\sin anglealpha \\sin anglebeta>0 .\n\\]\n\nTherefore\n\\[\n\\begin{array}{l}\nanglealpha \\cos anglebeta+\\sin anglealpha \\cos anglegamma \\geq \\tan anglealpha \\cos anglebeta+\\sin anglealpha \\cos anglegamma \\\\\n\\quad=\\tan anglealpha(\\cos anglebeta+\\cos anglealpha \\cos anglegamma)>0 .\n\\end{array}\n\\]\n\nThus the required inequality is proved in Case 2 also."
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "raincloud",
        "B": "stonegate",
        "C": "windchime"
      },
      "question": "<<<\n4. Prove that if \\( raincloud, stonegate \\), and \\( windchime \\) are angles of a triangle measured in radians then \\( raincloud \\cos stonegate+\\sin raincloud \\cos windchime>0 \\).\n>>>",
      "solution": "<<<\nSolution. We distinguish two cases.\nCase 1. \\( 0<stonegate<\\pi / 2 \\). Then \\( \\cos stonegate>0 \\). Since \\( raincloud>0, raincloud>\\sin raincloud \\); hence \\( raincloud \\cos stonegate>\\sin raincloud \\cos stonegate \\). Also \\( windchime=\\pi-raincloud-stonegate<\\pi-stonegate \\), and \\( 0<windchime<\\pi \\), so \\( \\cos windchime>\\cos (\\pi-stonegate)=-\\cos stonegate \\). Therefore \\( \\cos stonegate+ \\) \\( \\cos windchime>0 \\). Hence\n\\[\nraincloud \\cos stonegate+\\sin raincloud \\cos windchime>\\sin raincloud(\\cos stonegate+\\cos windchime)>0\n\\]\nand the desired inequality is established in Case 1.\nCase 2. \\( \\frac{\\pi}{2} \\leq stonegate<\\pi \\). Then \\( \\cos stonegate \\leq 0,0<raincloud<\\frac{\\pi}{2} \\), so \\( raincloud<\\tan raincloud \\), and \\( \\tan raincloud>0 \\). Hence\n\\[\nraincloud \\cos stonegate \\geq \\tan raincloud \\cos stonegate .\n\\]\n\nAlso \\( stonegate=\\pi-raincloud-windchime \\). So\n\\[\n\\cos stonegate+\\cos raincloud \\cos windchime=-\\cos (raincloud+windchime)+\\cos raincloud \\cos windchime=\\sin raincloud \\sin stonegate>0 .\n\\]\n\nTherefore\n\\[\n\\begin{array}{l}\nraincloud \\cos stonegate+\\sin raincloud \\cos windchime \\geq \\tan raincloud \\cos stonegate+\\sin raincloud \\cos windchime \\\\\n\\quad=\\tan raincloud(\\cos stonegate+\\cos raincloud \\cos windchime)>0 .\n\\end{array}\n\\]\n\nThus the required inequality is proved in Case 2 also.\n>>>"
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "straightness",
        "B": "flatness",
        "C": "linearity"
      },
      "question": "4. Prove that if \\( straightness, flatness \\), and \\( linearity \\) are angles of a triangle measured in radians then \\( straightness \\cos flatness+\\sin straightness \\cos linearity>0 \\).",
      "solution": "Solution. We distinguish two cases.\nCase 1. \\( 0<flatness<\\pi / 2 \\). Then \\( \\cos flatness>0 \\). Since \\( straightness>0, straightness>\\sin straightness \\); hence \\( straightness \\cos flatness>\\sin straightness \\cos flatness \\). Also \\( linearity=\\pi-straightness-flatness<\\pi-flatness \\), and \\( 0<linearity<\\pi \\), so \\( \\cos linearity>\\cos (\\pi-flatness)=-\\cos flatness \\). Therefore \\( \\cos flatness+ \\cos linearity>0 \\). Hence\n\\[\nstraightness \\cos flatness+\\sin straightness \\cos linearity>\\sin straightness(\\cos flatness+\\cos linearity)>0\n\\]\nand the desired inequality is established in Case 1.\nCase 2. \\( \\frac{\\pi}{2} \\leq flatness<\\pi \\). Then \\( \\cos flatness \\leq 0,0<straightness<\\frac{\\pi}{2} \\), so \\( straightness<\\tan straightness \\), and \\( \\tan straightness>0 \\). Hence\n\\[\nstraightness \\cos flatness \\geq \\tan straightness \\cos flatness .\n\\]\n\nAlso \\( flatness=\\pi-straightness-linearity \\). So\n\\[\n\\cos flatness+\\cos straightness \\cos linearity=-\\cos (straightness+linearity)+\\cos straightness \\cos linearity=\\sin straightness \\sin flatness>0 .\n\\]\n\nTherefore\n\\[\n\\begin{array}{l}\nstraightness \\cos flatness+\\sin straightness \\cos linearity \\geq \\tan straightness \\cos flatness+\\sin straightness \\cos linearity \\\\\n\\quad=\\tan straightness(\\cos flatness+\\cos straightness \\cos linearity)>0 .\n\\end{array}\n\\]\n\nThus the required inequality is proved in Case 2 also."
    },
    "garbled_string": {
      "map": {
        "A": "qzxwvtnp",
        "B": "hjgrksla",
        "C": "mncbdjfo"
      },
      "question": "4. Prove that if \\( qzxwvtnp, hjgrksla \\), and \\( mncbdjfo \\) are angles of a triangle measured in radians then \\( qzxwvtnp \\cos hjgrksla+\\sin qzxwvtnp \\cos mncbdjfo>0 \\).",
      "solution": "Solution. We distinguish two cases.\nCase 1. \\( 0<hjgrksla<\\pi / 2 \\). Then \\( \\cos hjgrksla>0 \\). Since \\( qzxwvtnp>0, qzxwvtnp>\\sin qzxwvtnp \\); hence \\( qzxwvtnp \\cos hjgrksla>\\sin qzxwvtnp \\cos hjgrksla \\). Also \\( mncbdjfo=\\pi-qzxwvtnp-hjgrksla<\\pi-hjgrksla \\), and \\( 0<mncbdjfo<\\pi \\), so \\( \\cos mncbdjfo>\\cos (\\pi-hjgrksla)=-\\cos hjgrksla \\). Therefore \\( \\cos hjgrksla+ \\) \\( \\cos mncbdjfo>0 \\). Hence\n\\[\nqzxwvtnp \\cos hjgrksla+\\sin qzxwvtnp \\cos mncbdjfo>\\sin qzxwvtnp(\\cos hjgrksla+\\cos mncbdjfo)>0\n\\]\nand the desired inequality is established in Case 1.\nCase 2. \\( \\frac{\\pi}{2} \\leq hjgrksla<\\pi \\). Then \\( \\cos hjgrksla \\leq 0,0<qzxwvtnp<\\frac{\\pi}{2} \\), so \\( qzxwvtnp<\\tan qzxwvtnp \\), and \\( \\tan qzxwvtnp>0 \\). Hence\n\\[\nqzxwvtnp \\cos hjgrksla \\geq \\tan qzxwvtnp \\cos hjgrksla .\n\\]\n\nAlso \\( hjgrksla=\\pi-qzxwvtnp-mncbdjfo \\). So\n\\[\n\\cos hjgrksla+\\cos qzxwvtnp \\cos mncbdjfo=-\\cos (qzxwvtnp+mncbdjfo)+\\cos qzxwvtnp \\cos mncbdjfo=\\sin qzxwvtnp \\sin hjgrksla>0 .\n\\]\n\nTherefore\n\\[\n\\begin{array}{l}\nqzxwvtnp \\cos hjgrksla+\\sin qzxwvtnp \\cos mncbdjfo \\geq \\tan qzxwvtnp \\cos hjgrksla+\\sin qzxwvtnp \\cos mncbdjfo \\\\\n\\quad=\\tan qzxwvtnp(\\cos hjgrksla+\\cos qzxwvtnp \\cos mncbdjfo)>0 .\n\\end{array}\n\\]\n\nThus the required inequality is proved in Case 2 also."
    },
    "kernel_variant": {
      "question": "Let A, B, and C be the angles (measured in radians) of a triangle, so that A, B, C are positive and A + B + C = π.  Prove the inequality\n\n      A·cos B + sin A·cos C > 0.",
      "solution": "Because A, B, C are angles of a (Euclidean) triangle we have 0 < A, B, C < π and A + B + C = π.\n\nWe split the proof according to the sign of cos B.\n\n────────────────────────────────────────────────────────\nCase 1: cos B ≥ 0   (that is, 0 < B ≤ π/2).\n────────────────────────────────────────────────────────\n\nStep 1.  Compare A and sin A.\nFor every x in (0, π) one has x > sin x; hence\n      A ≥ sin A and therefore A·cos B ≥ sin A·cos B.     (1)\n(The inequality is strict unless B = π/2.)\n\nStep 2.  Show cos B + cos C is positive.\nSince C = π − A − B < π − B, we have 0 < C < π and\n      cos C > cos(π − B) = −cos B.\nThus\n      cos B + cos C > 0.                                (2)\n\nStep 3.  Combine (1) and (2).\nUsing (1) and (2) we get\n  A·cos B + sin A·cos C\n     ≥ sin A·cos B + sin A·cos C\n     = sin A (cos B + cos C)\n     > 0.\n\n(In the boundary case B = π/2 one has cos B = 0 and the left–hand side equals sin A·cos C = sin A·sin A = sin²A > 0, so the conclusion still holds.)\n\n────────────────────────────────────────────────────────\nCase 2: cos B < 0   (that is, π/2 < B < π).\n────────────────────────────────────────────────────────\n\nStep 1.  Compare A and tan A.\nBecause 0 < A < π/2, the function x ↦ tan x exceeds x, so A < tan A.  Multiplying by the negative quantity cos B reverses the inequality:\n      A·cos B ≥ tan A·cos B.                            (3)\n\nStep 2.  Rewrite cos B in terms of A and C.\nFrom B = π − A − C we obtain\n      cos B + cos A·cos C\n      = −cos(A + C) + cos A·cos C\n      = −(cos A·cos C − sin A·sin C) + cos A·cos C\n      = sin A·sin C > 0.                               (4)\n\nStep 3.  Combine (3) and (4).\nSince sin A·cos C = tan A · (cos A·cos C),\n\ntan A·cos B + sin A·cos C\n      = tan A (cos B + cos A·cos C)    (by definition)\n      > 0  (by (4) and tan A > 0).\n\nUsing (3) we finally get\n  A·cos B + sin A·cos C\n     ≥ tan A·cos B + sin A·cos C > 0.\n\n────────────────────────────────────────────────────────\nConclusion.\nIn both cases the expression A·cos B + sin A·cos C is strictly positive; therefore\n      A·cos B + sin A·cos C > 0,\nas was to be proved.  ■",
      "_meta": {
        "core_steps": [
          "Partition on the sign of cos B (i.e. B<π/2 vs. B≥π/2).",
          "For an acute angle x, use the elementary bounds  x>sin x  and  x<tan x.",
          "Exploit the triangle relation C=π−A−B to compare cosines:  (i) cos C>−cos B when cos B>0;  (ii) cos B+cos A cos C=sin A sin B>0 when cos B≤0.",
          "Combine the above inequalities so that A cos B+sin A cos C is bounded below by a product of positive quantities in each case.",
          "Conclude A cos B+sin A cos C>0 for all triangle angles."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "The actual phrasing of the case split may be given as ‘cos B>0 / cos B≤0’ instead of ‘0<B<π/2 / π/2≤B<π’. The reasoning is unchanged because both descriptions capture the sign of cos B.",
            "original": "Case 1: 0<B<π/2; Case 2: π/2≤B<π"
          },
          "slot2": {
            "description": "The goal could be stated with ≥ instead of >; the proof still delivers the stronger ‘>’ so the chain of arguments is unaffected.",
            "original": "Prove A cos B + sin A cos C > 0"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}