인공지능(AI) 수학능력 TEST

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• • 
  • 세상의모든계산기 (*.105.205.197) 2024.05.21 10:41 #comment_40794

     

    ChatGPT-4o 에게 물어 봤습니다. 

    ---

    To determine if the equation 

    $-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right) = -\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

    is true or false, we can simplify and compare both sides. 

     

    ### Left Side:

    $-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right)$

    ### Right Side:

    $-\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

     

    ### Using the Identity for $\arctan$:

     

    We use the identity for the sum of arctangents:

    $\arctan(a) + \arctan(b) = \arctan\left(\frac{a + b}{1 - ab}\right)$

     

    This identity holds when $ab < 1$.
    In this case, let $a = \frac{11}{5\sqrt{3}}$ and $b = \frac{5\sqrt{3}}{11}$:

     

    $a \cdot b = \left(\frac{11}{5\sqrt{3}}\right) \cdot \left(\frac{5\sqrt{3}}{11}\right) = 1$

     

    Since $ab = 1$, the sum formula becomes:

     

    $\arctan\left(\frac{11}{5\sqrt{3}}\right) + \arctan\left(\frac{5\sqrt{3}}{11}\right) = \frac{\pi}{2}$

     

    Thus,

     

    $\arctan\left(\frac{11}{5\sqrt{3}}\right) = \frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

     

    Now, substitute this back into the original equation:

     

    ### Left Side Simplification:

     

    $-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right) = -\pi + \left(\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)\right)$

     

    $= -\pi + \frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

     

    $= -\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

     

    ### Comparing Left and Right Sides:

     

    The simplified left side is:

     

    $-\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

     

    This matches the right side exactly. Therefore, the original equation is:

     

    $-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right) = -\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)$

     

    Thus, the statement is **true**.

     
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• • 조니 (*.113.121.69) 2024.05.22 21:15 #comment_40848

    저도 chatgpt-4인데 false가 나오네요. 이 것도 부호 계산을 틀리게 하네요. ㅠㅠ

    

     

     

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• • 조니 (*.113.121.69) 2024.05.22 22:30 #comment_40871

    

     

    chatgpt 3.5는 true가 나오는게 approximate를 사용해서 같다고 결론을 도출하는데, 이건 아닌거 같습니다. ㅎㅎ

    그리고 한글 버전에서는 계산이 안되는데 영문 버전에서는 그나마 엉터리 같지만 결과가 근사하게 나오기는 하네요.

     

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• • 조니 (*.113.121.69) 2024.05.23 21:33 #comment_40886

    chatgpt-4o는 동일하게 true 나오네요.

    

     

    ㅊ

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    arctan_error.JPG 127.5KB / 18
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• • 
  • 세상의모든계산기 (*.105.205.197) 2024.05.21 10:44 #comment_40799

    제미나이에게도 같은 질문으로 물어 봤습니다. 

    ---

    The statement "-Pi + ArcTan[11/(5 Sqrt[3])] == -Pi/2 - ArcTan[(5 Sqrt[3])/11]" is false.

    To prove this, we can simplify both sides of the equation:

    Left side:

    -Pi + ArcTan[11/(5 Sqrt[3])]

    = -Pi + ArcTan[11/(5 * 3^(1/2))]

    = -Pi + ArcTan[11/(5 * (3^1/2))]

    = -Pi + ArcTan[(11 * 3^(-1/2)) / 5]

    = -Pi + ArcTan[(11 * (1/3^(1/2))) / 5]

    = -Pi + ArcTan[(11/3^(1/2)) / 5]

    Right side:

    -Pi/2 - ArcTan[(5 Sqrt[3])/11]

    = -Pi/2 - ArcTan[(5 * 3^(1/2)) / 11]

    = -Pi/2 - ArcTan[(5 * (3^1/2)) / 11]

    = -Pi/2 - ArcTan[(5/11) * (3^1/2)]

    = -Pi/2 - ArcTan[(5/11) * (1/3^(1/2))]

    = -Pi/2 - ArcTan[(5/(11 * 3^(1/2)))]

    = -Pi/2 - ArcTan[(5/(11 * (1/3^(1/2)))]

    = -Pi/2 - ArcTan[(5/(11/3^(1/2)))]

    = -Pi/2 - ArcTan[(5 * 3^(1/2)) / 11]

    Comparing the simplified forms of both sides, we see that they are not equal. Therefore, the original statement is false.

    Additional verification:

    We can also use a graphing calculator to verify that the two expressions do not have the same value. Plotting both expressions on a graph, we can see that they have different y-intercepts and slopes. This confirms that the two expressions are not equal.

    Conclusion:

    Based on the above analysis, we can conclude that the statement "-Pi + ArcTan[11/(5 Sqrt[3])] == -Pi/2 - ArcTan[(5 Sqrt[3])/11]" is false.

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• • 조니 (*.113.121.69) 2024.05.22 21:03 #comment_40828

    제미나이 저도 해 봤는데 true라고 나오네요

    그런데 이건 좀 희한한게 tan(pi/6)을 -루트3/3으로 부호를 틀리게 계산을 했는데도 결과가 맞다고 도출하는게 올바른 결과인지 의문입니다.
    

     

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• • 
  • 세상의모든계산기 (*.105.205.197) 2024.05.21 10:56 #comment_40802

    Copilot 에게도 물어 봤습니다.

    ---

    
    ​​​True! The given equation holds. Both sides are equal. Here’s the breakdown:

    1. Start with the left-hand side (LHS):

    $-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right)$

    2. Simplify the fraction:

    $\frac{11}{5\sqrt{3}} = \frac{11\sqrt{3}}{15}$

    3. Use the identity:

    $\arctan(x) + \arctan(1/x) = \frac{\pi}{2}$

    Applying this to our expression:

    $\arctan\left(\frac{11\sqrt{3}}{15}\right) + \arctan\left(\frac{15}{11\sqrt{3}}\right) = \frac{\pi}{2}$

    4. Simplify the second term:

    $\frac{15}{11\sqrt{3}} = \frac{5\sqrt{3}}{11}$

    5. Combine the arctan terms:

    $\arctan\left(\frac{11\sqrt{3}}{15}\right) + \arctan\left(\frac{5\sqrt{3}}{11}\right) = \frac{\pi}{2}$

    6. Use the identity again:

    $\arctan(x) + \arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$

    Applying this to our expression:

    $\arctan\left(\frac{11\sqrt{3} + 5\sqrt{3}}{1 - (11\sqrt{3})(5\sqrt{3})}\right) = \frac{\pi}{2}$

    7. Simplify the numerator:

    $11\sqrt{3} + 5\sqrt{3} = 16\sqrt{3}$

    And the denominator:

    $1 - (11\sqrt{3})(5\sqrt{3}) = 1 - 165 = -164$

    8. Final result:

    $\arctan\left(\frac{16\sqrt{3}}{-164}\right) = \frac{\pi}{2}$

    Since $\frac{16\sqrt{3}}{-164} = \frac{\sqrt{3}}{-41}$

    , we have:

    $\arctan\left(\frac{\sqrt{3}}{-41}\right) = \frac{\pi}{2}$ Therefore, the LHS equals $-\pi/2$

    .

    9. Now let’s look at the right-hand side (RHS):

    $-\pi/2 - \arctan\left(\frac{5\sqrt{3}} {11}\right)$

    10. We already found that $\arctan\left(\frac{5\sqrt{3}}{11}\right) = \frac{\pi}{2}$

    So the RHS also equals $-\pi/2$

    .

    Since both sides are equal, the given equation is true! �

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• • 조니 (*.113.121.69) 2024.05.22 21:37 #comment_40861
    copilot은 한달 후에 유료라고 해서 안 해봤습니다.
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• • 
  • 세상의모든계산기 (*.105.205.197) 2024.05.23 09:14 #comment_40880
    AI녀석은
    "그때 그때 달라요"
    인가보네요.
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