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人気のある三角関数 >

solvefor x,y=(sec(x))/(cos^1(x))+8y^2

解

解く

x, y = \frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2}

解

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn, x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

解答ステップ

y = \frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2}

辺を交換する

\frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2} = y

両辺から

y

を引く

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y = 0

簡素化

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y : \frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )}

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y

元を分数に変換する:

8 y^{2} = \frac{8 y ^{2} cos ( x )}{cos ( x )}, y = \frac{y cos ( x )}{cos ( x )}

= \frac{sec ( x )}{cos ( x )} + \frac{8 y ^{2} cos ( x )}{cos ( x )} - \frac{y cos ( x )}{cos ( x )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )}

\frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sec (x) + 8 y^{2} cos (x) - y cos (x) = 0

三角関数の公式を使用して書き換える

sec (x) - cos (x) y + 8 cos (x) y^{2}

基本的な三角関数の公式を使用する:

cos (x) = \frac{1}{sec ( x )}

= sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

簡素化

sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2} : sec (x) + \frac{- y + 8 y ^{2}}{sec ( x )}

sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

\frac{1}{sec ( x )} y = \frac{y}{sec ( x )}

\frac{1}{sec ( x )} y

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot y}{sec ( x )}

乗算:

1 \cdot y = y

= \frac{y}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} y^{2} = \frac{8 y ^{2}}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} y^{2}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 8 y ^{2}}{sec ( x )}

数を乗じる:

1 \cdot 8 = 8

= \frac{8 y ^{2}}{sec ( x )}

= sec (x) - \frac{y}{sec ( x )} + \frac{8 y ^{2}}{sec ( x )}

分数を組み合わせる

- \frac{y}{sec ( x )} + \frac{8 y ^{2}}{sec ( x )} : \frac{- y + 8 y ^{2}}{sec ( x )}

規則を適用

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- y + 8 y ^{2}}{sec ( x )}

= sec (x) + \frac{8 y ^{2} - y}{sec ( x )}

= sec (x) + \frac{- y + 8 y ^{2}}{sec ( x )}

\frac{- y + 8 y ^{2}}{sec ( x )} + sec (x) = 0

置換で解く

\frac{- y + 8 y ^{2}}{sec ( x )} + sec (x) = 0

仮定:

sec (x) = u

\frac{- y + 8 y ^{2}}{u} + u = 0

\frac{- y + 8 y ^{2}}{u} + u = 0 : u = y - 8 y^{2}, u = - y - 8 y^{2}

\frac{- y + 8 y ^{2}}{u} + u = 0

以下で両辺を乗じる:

u

\frac{- y + 8 y ^{2}}{u} + u = 0

以下で両辺を乗じる:

u

\frac{- y + 8 y ^{2}}{u} u + uu = 0 \cdot u

簡素化

\frac{- y + 8 y ^{2}}{u} u + uu = 0 \cdot u

簡素化

\frac{- y + 8 y ^{2}}{u} u : - y + 8 y^{2}

\frac{- y + 8 y ^{2}}{u} u

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( - y + 8 y ^{2} ) u}{u}

共通因数を約分する:

u

= - - y + 8 y^{2}

簡素化

uu : u^{2}

uu

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

数を足す:

1 + 1 = 2

= u^{2}

簡素化

0 \cdot u : 0

0 \cdot u

規則を適用

0 \cdot a = 0

= 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

解く

- y + 8 y^{2} + u^{2} = 0 : u = y - 8 y^{2}, u = - y - 8 y^{2}

- y + 8 y^{2} + u^{2} = 0

y

を右側に移動します

- y + 8 y^{2} + u^{2} = 0

両辺に

y

を足す

- y + 8 y^{2} + u^{2} + y = 0 + y

簡素化

8 y^{2} + u^{2} = y

8 y^{2} + u^{2} = y

8 y^{2}

を右側に移動します

8 y^{2} + u^{2} = y

両辺から

8 y^{2}

を引く

8 y^{2} + u^{2} - 8 y^{2} = y - 8 y^{2}

簡素化

u^{2} = y - 8 y^{2}

u^{2} = y - 8 y^{2}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = y - 8 y^{2}, u = - y - 8 y^{2}

u = y - 8 y^{2}, u = - y - 8 y^{2}

代用を戻す

u = sec (x)

sec (x) = y - 8 y^{2}, sec (x) = - y - 8 y^{2}

sec (x) = y - 8 y^{2}, sec (x) = - y - 8 y^{2}

sec (x) = y - 8 y^{2} : x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

sec (x) = y - 8 y^{2}

三角関数の逆数プロパティを適用する

sec (x) = y - 8 y^{2}

以下の一般解

sec (x) = y - 8 y^{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

sec (x) = - y - 8 y^{2} : x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

sec (x) = - y - 8 y^{2}

三角関数の逆数プロパティを適用する

sec (x) = - y - 8 y^{2}

以下の一般解

sec (x) = - y - 8 y^{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

すべての解を組み合わせる

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn, x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

グラフ

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