인공지능(AI) 수학능력 TEST

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  세상의모든계산기

  2024.05.21 - 10:41 #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

  2024.05.22 - 22:28 2024.05.22 - 21:15 #40848

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

  

   

   

  Attached file

  chatgpt-4_false.JPG 110.0KB 36

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  조니

  2024.05.22 - 22:31 2024.05.22 - 22:30 #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.

   

  Attached file

  chatgpt-35.JPG 39.7KB 37

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  조니

  2024.05.23 - 21:33 #40886

  chatgpt-4o는 동일하게 true 나오네요.

  

   

  ㅊ

  Attached file

  arctan_error.JPG 127.5KB 32

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  세상의모든계산기

  2024.05.21 - 10:44 #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

  2024.05.22 - 21:29 2024.05.22 - 21:03 #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.
  

   

  Attached file

  chatgpt-4_true_gemini.JPG 78.6KB 43

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  세상의모든계산기

  2024.05.21 - 11:02 2024.05.21 - 10:56 #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

  2024.05.22 - 21:37 #40861

  copilot은 한달 후에 유료라고 해서 안 해봤습니다.

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  세상의모든계산기

  2024.05.23 - 09:14 #40880

  AI녀석은
  "그때 그때 달라요"
  인가보네요.

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