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人気のある三角関数 >

sin^3(o)=4sin(o)sin^2(o)sin^4(o)

解

sin^{3} (o) = 4 sin (o) sin^{2} (o) sin^{4} (o)

解

o = 2 πn, o = π + 2 πn, o = \frac{5 π}{4} + 2 πn, o = \frac{7 π}{4} + 2 πn, o = \frac{π}{4} + 2 πn, o = \frac{3 π}{4} + 2 πn

+1

度

o = 0^{\circ} + 36 0^{\circ} n, o = 18 0^{\circ} + 36 0^{\circ} n, o = 22 5^{\circ} + 36 0^{\circ} n, o = 31 5^{\circ} + 36 0^{\circ} n, o = 4 5^{\circ} + 36 0^{\circ} n, o = 13 5^{\circ} + 36 0^{\circ} n

解答ステップ

sin^{3} (o) = 4 sin (o) sin^{2} (o) sin^{4} (o)

置換で解く

sin^{3} (o) = 4 sin (o) sin^{2} (o) sin^{4} (o)

仮定:

sin (o) = u

u^{3} = 4 u u^{2} u^{4}

u^{3} = 4 u u^{2} u^{4} : u = 0, u = i \frac{1}{2}, u = - i \frac{1}{2}, u = - \frac{2}{2}, u = \frac{2}{2}

u^{3} = 4 u u^{2} u^{4}

辺を交換する

4 u u^{2} u^{4} = u^{3}

u^{3}

を左側に移動します

4 u u^{2} u^{4} = u^{3}

両辺から

u^{3}

を引く

4 u u^{2} u^{4} - u^{3} = u^{3} - u^{3}

簡素化

4 u^{7} - u^{3} = 0

4 u^{7} - u^{3} = 0

因数

4 u^{7} - u^{3} : u^{3} (2 u^{2} + 1) (2 u + 1) (2 u - 1)

4 u^{7} - u^{3}

共通項をくくり出す

u^{3} : u^{3} (4 u^{4} - 1)

4 u^{7} - u^{3}

指数の規則を適用する:

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

u^{7} = u^{4} u^{3}

= 4 u^{4} u^{3} - u^{3}

共通項をくくり出す

u^{3}

= u^{3} (4 u^{4} - 1)

= u^{3} (4 u^{4} - 1)

因数

4 u^{4} - 1 : (2 u^{2} + 1) (2 u + 1) (2 u - 1)

4 u^{4} - 1

4 u^{4} - 1

を書き換え

(2 u^{2})^{2} - 1^{2}

4 u^{4} - 1

4

を書き換え

2^{2}

= 2^{2} u^{4} - 1

1

を書き換え

1^{2}

= 2^{2} u^{4} - 1^{2}

指数の規則を適用する:

a^{b c} = (a^{b})^{c}

u^{4} = (u^{2})^{2}

= 2^{2} (u^{2})^{2} - 1^{2}

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

2^{2} (u^{2})^{2} = (2 u^{2})^{2}

= (2 u^{2})^{2} - 1^{2}

= (2 u^{2})^{2} - 1^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(2 u^{2})^{2} - 1^{2} = (2 u^{2} + 1) (2 u^{2} - 1)

= (2 u^{2} + 1) (2 u^{2} - 1)

因数

2 u^{2} - 1 : (2 u + 1) (2 u - 1)

2 u^{2} - 1

2 u^{2} - 1

を書き換え

(2 u)^{2} - 1^{2}

2 u^{2} - 1

累乗根の規則を適用する:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} u^{2} - 1

1

を書き換え

1^{2}

= (2)^{2} u^{2} - 1^{2}

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

(2)^{2} u^{2} = (2 u)^{2}

= (2 u)^{2} - 1^{2}

= (2 u)^{2} - 1^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(2 u)^{2} - 1^{2} = (2 u + 1) (2 u - 1)

= (2 u + 1) (2 u - 1)

= (2 u^{2} + 1) (2 u + 1) (2 u - 1)

= u^{3} (2 u^{2} + 1) (2 u + 1) (2 u - 1)

u^{3} (2 u^{2} + 1) (2 u + 1) (2 u - 1) = 0

零因子の原則を使用:

ab = 0

ならば

a = 0

または

b = 0

u = 0 or 2 u^{2} + 1 = 0 or 2 u + 1 = 0 or 2 u - 1 = 0

解く

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

2 u^{2} + 1 = 0

1

を右側に移動します

2 u^{2} + 1 = 0

両辺から

1

を引く

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

簡素化

2 u^{2} = - 1

2 u^{2} = - 1

以下で両辺を割る

2

2 u^{2} = - 1

以下で両辺を割る

2

\frac{2 u ^{2}}{2} = \frac{- 1}{2}

簡素化

u^{2} = - \frac{1}{2}

u^{2} = - \frac{1}{2}

x^{2} = f (a)

の場合, 解は

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

u = - \frac{1}{2}, u = - - \frac{1}{2}

簡素化

- \frac{1}{2} : i \frac{1}{2}

- \frac{1}{2}

累乗根の規則を適用する:

- a = - 1 a

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

= - 1 \frac{1}{2}

虚数の規則を適用する:

- 1 = i

= i \frac{1}{2}

簡素化

- - \frac{1}{2} : - i \frac{1}{2}

- - \frac{1}{2}

簡素化

- \frac{1}{2} : i \frac{1}{2}

- \frac{1}{2}

累乗根の規則を適用する:

- a = - 1 a

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

= - 1 \frac{1}{2}

虚数の規則を適用する:

- 1 = i

= i \frac{1}{2}

= - i \frac{1}{2}

u = i \frac{1}{2}, u = - i \frac{1}{2}

解く

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

2 u + 1 = 0

1

を右側に移動します

2 u + 1 = 0

両辺から

1

を引く

2 u + 1 - 1 = 0 - 1

簡素化

2 u = - 1

2 u = - 1

以下で両辺を割る

2

2 u = - 1

以下で両辺を割る

2

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

簡素化

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

簡素化

\frac{2 u}{2} : u

\frac{2 u}{2}

共通因数を約分する:

2

= u

簡素化

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

分数の規則を適用する:

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

= - \frac{1}{2}

有理化する

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

共役で乗じる

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

累乗根の規則を適用する:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

解く

2 u - 1 = 0 : u = \frac{2}{2}

2 u - 1 = 0

1

を右側に移動します

2 u - 1 = 0

両辺に

1

を足す

2 u - 1 + 1 = 0 + 1

簡素化

2 u = 1

2 u = 1

以下で両辺を割る

2

2 u = 1

以下で両辺を割る

2

\frac{2 u}{2} = \frac{1}{2}

簡素化

\frac{2 u}{2} = \frac{1}{2}

簡素化

\frac{2 u}{2} : u

\frac{2 u}{2}

共通因数を約分する:

2

= u

簡素化

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

共役で乗じる

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

累乗根の規則を適用する:

a a = a

22 = 2

= 2

= \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

解答は

u = 0, u = i \frac{1}{2}, u = - i \frac{1}{2}, u = - \frac{2}{2}, u = \frac{2}{2}

代用を戻す

u = sin (o)

sin (o) = 0, sin (o) = i \frac{1}{2}, sin (o) = - i \frac{1}{2}, sin (o) = - \frac{2}{2}, sin (o) = \frac{2}{2}

sin (o) = 0, sin (o) = i \frac{1}{2}, sin (o) = - i \frac{1}{2}, sin (o) = - \frac{2}{2}, sin (o) = \frac{2}{2}

sin (o) = 0 : o = 2 πn, o = π + 2 πn

sin (o) = 0

以下の一般解

sin (o) = 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}

o = 0 + 2 πn, o = π + 2 πn

o = 0 + 2 πn, o = π + 2 πn

解く

o = 0 + 2 πn : o = 2 πn

o = 0 + 2 πn

0 + 2 πn = 2 πn

o = 2 πn

o = 2 πn, o = π + 2 πn

sin (o) = i \frac{1}{2} :

解なし

sin (o) = i \frac{1}{2}

解なし

sin (o) = - i \frac{1}{2} :

解なし

sin (o) = - i \frac{1}{2}

解なし

sin (o) = - \frac{2}{2} : o = \frac{5 π}{4} + 2 πn, o = \frac{7 π}{4} + 2 πn

sin (o) = - \frac{2}{2}

以下の一般解

sin (o) = - \frac{2}{2}

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}

o = \frac{5 π}{4} + 2 πn, o = \frac{7 π}{4} + 2 πn

o = \frac{5 π}{4} + 2 πn, o = \frac{7 π}{4} + 2 πn

sin (o) = \frac{2}{2} : o = \frac{π}{4} + 2 πn, o = \frac{3 π}{4} + 2 πn

sin (o) = \frac{2}{2}

以下の一般解

sin (o) = \frac{2}{2}

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}

o = \frac{π}{4} + 2 πn, o = \frac{3 π}{4} + 2 πn

o = \frac{π}{4} + 2 πn, o = \frac{3 π}{4} + 2 πn

すべての解を組み合わせる

o = 2 πn, o = π + 2 πn, o = \frac{5 π}{4} + 2 πn, o = \frac{7 π}{4} + 2 πn, o = \frac{π}{4} + 2 πn, o = \frac{3 π}{4} + 2 πn

グラフ

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