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人気のある三角関数 >

sin(x)sin^3(x)-sin^5(x)sin^3(x)=0

解

sin (x) sin^{3} (x) - sin^{5} (x) sin^{3} (x) = 0

解

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

解答ステップ

sin (x) sin^{3} (x) - sin^{5} (x) sin^{3} (x) = 0

置換で解く

sin (x) sin^{3} (x) - sin^{5} (x) sin^{3} (x) = 0

仮定:

sin (x) = u

u u^{3} - u^{5} u^{3} = 0

u u^{3} - u^{5} u^{3} = 0 : u = 0, u = 1, u = - 1, u = i, u = - i

u u^{3} - u^{5} u^{3} = 0

改良

u^{4} - u^{8} = 0

標準的な形式で書く

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- u^{8} + u^{4} = 0

equationを

v = u^{2}, v^{2} = u^{4}

と以下で書き換える:

v^{4} = u^{8}

- v^{4} + v^{2} = 0

解く

- v^{4} + v^{2} = 0 : v = 0, v = 1, v = - 1

- v^{4} + v^{2} = 0

equationを

u = v^{2}

と以下で書き換える:

u^{2} = v^{4}

- u^{2} + u = 0

解く

- u^{2} + u = 0 : u = 0, u = 1

- u^{2} + u = 0

解くとthe二次式

- u^{2} + u = 0

二次Equationの公式:

次の場合:

a = - 1, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) \cdot 0}{2 ( - 1 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) \cdot 0}{2 ( - 1 )}

1^{2} - 4 (- 1) \cdot 0 = 1

1^{2} - 4 (- 1) \cdot 0

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 0

規則を適用

- (- a) = a

= 1 + 4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 1 + 0

数を足す:

1 + 0 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 ( - 1 )}

解を分離する

u_{1} = \frac{- 1 + 1}{2 ( - 1 )}, u_{2} = \frac{- 1 - 1}{2 ( - 1 )}

u = \frac{- 1 + 1}{2 ( - 1 )} : 0

\frac{- 1 + 1}{2 ( - 1 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 + 1}{- 2 \cdot 1}

数を足す/引く:

- 1 + 1 = 0

= \frac{0}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{0}{- 2}

分数の規則を適用する:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{0}{2}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 1 - 1}{2 ( - 1 )} : 1

\frac{- 1 - 1}{2 ( - 1 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 - 1}{- 2 \cdot 1}

数を引く:

- 1 - 1 = - 2

= \frac{- 2}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

分数の規則を適用する:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

u = 0, u = 1

u = 0, u = 1

再び

u = v^{2}

に置き換えて以下を解く:

v

解く

v^{2} = 0 : v = 0

v^{2} = 0

規則を適用

x^{n} = 0 \Rightarrow x = 0

v = 0

解く

v^{2} = 1 : v = 1, v = - 1

v^{2} = 1

x^{2} = f (a)

の場合, 解は

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

v = 1, v = - 1

1 = 1

1

規則を適用

1 = 1

= 1

- 1 = - 1

- 1

規則を適用

1 = 1

= - 1

v = 1, v = - 1

解答は

v = 0, v = 1, v = - 1

v = 0, v = 1, v = - 1

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

u^{2} = 0 : u = 0

u^{2} = 0

規則を適用

x^{n} = 0 \Rightarrow x = 0

u = 0

解く

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

x^{2} = f (a)

の場合, 解は

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

u = 1, u = - 1

1 = 1

1

規則を適用

1 = 1

= 1

- 1 = - 1

- 1

規則を適用

1 = 1

= - 1

u = 1, u = - 1

解く

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

x^{2} = f (a)

の場合, 解は

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

u = - 1, u = - - 1

簡素化

- 1 : i

- 1

虚数の規則を適用する:

- 1 = i

= i

簡素化

- - 1 : - i

- - 1

虚数の規則を適用する:

- 1 = i

= - i

u = i, u = - i

解答は

u = 0, u = 1, u = - 1, u = i, u = - i

代用を戻す

u = sin (x)

sin (x) = 0, sin (x) = 1, sin (x) = - 1, sin (x) = i, sin (x) = - i

sin (x) = 0, sin (x) = 1, sin (x) = - 1, sin (x) = i, sin (x) = - i

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

以下の一般解

sin (x) = 0

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

以下の一般解

sin (x) = 1

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

以下の一般解

sin (x) = - 1

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = i :

解なし

sin (x) = i

解なし

sin (x) = - i :

解なし

sin (x) = - i

解なし

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

グラフ

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