ChatGPT-4o 에게 물어 봤습니다.
To determine if the equation
-π+arctan1153=-π2-arctan5311-\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.
-π+arctan1153-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right)
-π2-arctan5311-\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)
We use the identity for the sum of arctangents:
arctan(a)+arctan(b)=arctana+b1-ab\arctan(a) + \arctan(b) = \arctan\left(\frac{a + b}{1 - ab}\right)
This identity holds when ab<1ab < 1. In this case, let a=1153a = \frac{11}{5\sqrt{3}} and b=5311b = \frac{5\sqrt{3}}{11}:
a·b=1153·5311=1a \cdot b = \left(\frac{11}{5\sqrt{3}}\right) \cdot \left(\frac{5\sqrt{3}}{11}\right) = 1
Since ab=1ab = 1, the sum formula becomes:
arctan1153+arctan5311=π2\arctan\left(\frac{11}{5\sqrt{3}}\right) + \arctan\left(\frac{5\sqrt{3}}{11}\right) = \frac{\pi}{2}
Thus,
arctan1153=π2-arctan5311\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:
-π+arctan1153=-π+π2-arctan5311-\pi + \arctan\left(\frac{11}{5\sqrt{3}}\right) = -\pi + \left(\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)\right)
=-π+π2-arctan5311= -\pi + \frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)
=-π2-arctan5311= -\frac{\pi}{2} - \arctan\left(\frac{5\sqrt{3}}{11}\right)
The simplified left side is:
This matches the right side exactly. Therefore, the original equation is:
Thus, the statement is **true**.
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ChatGPT-4o 에게 물어 봤습니다.
To determine if the equation
is true or false, we can simplify and compare both sides.
### Left Side:
### Right Side:
### Using the Identity for $\arctan$:
We use the identity for the sum of arctangents:
This identity holds when .
In this case, let and :
Since , the sum formula becomes:
Thus,
Now, substitute this back into the original equation:
### Left Side Simplification:
### Comparing Left and Right Sides:
The simplified left side is:
This matches the right side exactly. Therefore, the original equation is:
Thus, the statement is **true**.