\( \arctan ( x + y ) = y ^ { 2 } + \frac { \pi } { 4 } , \quad ( 1,0 ) \)

To analyze the equation \( \arctan ( x + y ) = y ^ { 2 } + \frac { \pi } { 4 } \) with the point \( (1,0) \), we first substitute \( x = 1 \) and \( y = 0 \). This gives us \( \arctan(1 + 0) = \arctan(1) = \frac{\pi}{4} \). On the right side, we have \( 0^2 + \frac{\pi}{4} = \frac{\pi}{4} \). Therefore, the point \( (1,0) \) satisfies the equation! Interestingly, this is a great example of how well-defined mathematical functions can help us visualize relationships between variables. The arctangent function often appears in calculus and trigonometry, often bridging real and complex analysis. On the problem-solving front, always remember to check your initial point against the equation to confirm it works. A common mistake would be to miscalculate \( \arctan \) values or misunderstand how they translate into the given equation. It's always a plus to sketch graphs or use numerical solutions for complex equations to make your understanding clearer!