solvefor y,six^2+y^2=z^2

解

解く

y, s i x^{2} + y^{2} = z^{2}

y = - \frac{s x ^{2}}{2 - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2}} + - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2} i, y = \frac{s x ^{2}}{2 - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2}} - - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2} i, y = - \frac{s x ^{2}}{2 - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}} + - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2} i, y = \frac{s x ^{2}}{2 - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}} - - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2} i

解答ステップ

s i x^{2} + y^{2} = z^{2}

代用

y = x + v i

s i x^{2} + (x + v i)^{2} = z^{2}

拡張

s i x^{2} + (x + v i)^{2} : (x^{2} - v^{2}) + i (s x^{2} + 2 xv)

x^{2} - v^{2} + i (s x^{2} + 2 xv) = z^{2}

標準的な複素数形式で

z^{2}

を書き換える:

z^{2} + 0 i

x^{2} - v^{2} + i (s x^{2} + 2 xv) = z^{2} + 0 i

複素数は, その実数部と虚数部が等しい場合にのみ等しくなるequation系として書き換える:

[x^{2} - v^{2} = z^{2} s x^{2} + 2 xv = 0]

[x^{2} - v^{2} = z^{2} s x^{2} + 2 xv = 0] : x = - \frac{s x ^{2}}{2 - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2}}, x = \frac{s x ^{2}}{2 - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2}}, x = - \frac{s x ^{2}}{2 - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}}, x = \frac{s x ^{2}}{2 - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}}, v = - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2} v = - - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2} v = - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2} v = - - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}

代用を戻す

y = x + v i

y = - \frac{s x ^{2}}{2 - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2}} + - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2} i, y = \frac{s x ^{2}}{2 - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2}} - - \frac{z ^{2} + s ^{2} x ^{4} + z ^{4}}{2} i, y = - \frac{s x ^{2}}{2 - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}} + - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2} i, y = \frac{s x ^{2}}{2 - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2}} - - \frac{z ^{2} - s ^{2} x ^{4} + z ^{4}}{2} i