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人気のある三角関数 >

cot^2(x)csc^2(x)-cot^2(x)=9

解

cot^{2} (x) csc^{2} (x) - cot^{2} (x) = 9

解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

解答ステップ

cot^{2} (x) csc^{2} (x) - cot^{2} (x) = 9

両辺から

9

を引く

cot^{2} (x) csc^{2} (x) - cot^{2} (x) - 9 = 0

三角関数の公式を使用して書き換える

- 9 - cot^{2} (x) + cot^{2} (x) csc^{2} (x)

ピタゴラスの公式を使用する:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 9 - (csc^{2} (x) - 1) + (csc^{2} (x) - 1) csc^{2} (x)

簡素化

- 9 - (csc^{2} (x) - 1) + (csc^{2} (x) - 1) csc^{2} (x) : csc^{4} (x) - 2 csc^{2} (x) - 8

- 9 - (csc^{2} (x) - 1) + (csc^{2} (x) - 1) csc^{2} (x)

= - 9 - (csc^{2} (x) - 1) + csc^{2} (x) (csc^{2} (x) - 1)

- (csc^{2} (x) - 1) : - csc^{2} (x) + 1

- (csc^{2} (x) - 1)

括弧を分配する

= - (csc^{2} (x)) - (- 1)

マイナス・プラスの規則を適用する

- (- a) = a, - (a) = - a

= - csc^{2} (x) + 1

= - 9 - csc^{2} (x) + 1 + (csc^{2} (x) - 1) csc^{2} (x)

拡張

csc^{2} (x) (csc^{2} (x) - 1) : csc^{4} (x) - csc^{2} (x)

csc^{2} (x) (csc^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = csc^{2} (x), b = csc^{2} (x), c = 1

= csc^{2} (x) csc^{2} (x) - csc^{2} (x) \cdot 1

= csc^{2} (x) csc^{2} (x) - 1 \cdot csc^{2} (x)

簡素化

csc^{2} (x) csc^{2} (x) - 1 \cdot csc^{2} (x) : csc^{4} (x) - csc^{2} (x)

csc^{2} (x) csc^{2} (x) - 1 \cdot csc^{2} (x)

csc^{2} (x) csc^{2} (x) = csc^{4} (x)

csc^{2} (x) csc^{2} (x)

指数の規則を適用する:

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

csc^{2} (x) csc^{2} (x) = csc^{2 + 2} (x)

= csc^{2 + 2} (x)

数を足す:

2 + 2 = 4

= csc^{4} (x)

1 \cdot csc^{2} (x) = csc^{2} (x)

1 \cdot csc^{2} (x)

乗算:

1 \cdot csc^{2} (x) = csc^{2} (x)

= csc^{2} (x)

= csc^{4} (x) - csc^{2} (x)

= csc^{4} (x) - csc^{2} (x)

= - 9 - csc^{2} (x) + 1 + csc^{4} (x) - csc^{2} (x)

簡素化

- 9 - csc^{2} (x) + 1 + csc^{4} (x) - csc^{2} (x) : csc^{4} (x) - 2 csc^{2} (x) - 8

- 9 - csc^{2} (x) + 1 + csc^{4} (x) - csc^{2} (x)

条件のようなグループ

= - csc^{2} (x) + csc^{4} (x) - csc^{2} (x) - 9 + 1

類似した元を足す:

- csc^{2} (x) - csc^{2} (x) = - 2 csc^{2} (x)

= - 2 csc^{2} (x) + csc^{4} (x) - 9 + 1

数を足す/引く:

- 9 + 1 = - 8

= csc^{4} (x) - 2 csc^{2} (x) - 8

= csc^{4} (x) - 2 csc^{2} (x) - 8

= csc^{4} (x) - 2 csc^{2} (x) - 8

- 8 + csc^{4} (x) - 2 csc^{2} (x) = 0

置換で解く

- 8 + csc^{4} (x) - 2 csc^{2} (x) = 0

仮定:

csc (x) = u

- 8 + u^{4} - 2 u^{2} = 0

- 8 + u^{4} - 2 u^{2} = 0 : u = 2, u = - 2, u = 2 i, u = - 2 i

- 8 + u^{4} - 2 u^{2} = 0

標準的な形式で書く

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

u^{4} - 2 u^{2} - 8 = 0

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

v^{2} - 2 v - 8 = 0

解く

v^{2} - 2 v - 8 = 0 : v = 4, v = - 2

v^{2} - 2 v - 8 = 0

解くとthe二次式

v^{2} - 2 v - 8 = 0

二次Equationの公式:

次の場合:

a = 1, b = - 2, c = - 8

v_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot ( - 8 )}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot ( - 8 )}{2 \cdot 1}

(- 2)^{2} - 4 \cdot 1 \cdot (- 8) = 6

(- 2)^{2} - 4 \cdot 1 \cdot (- 8)

規則を適用

- (- a) = a

= (- 2)^{2} + 4 \cdot 1 \cdot 8

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

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

= 2^{2} + 4 \cdot 1 \cdot 8

数を乗じる:

4 \cdot 1 \cdot 8 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数を足す:

4 + 32 = 36

= 36

数を因数に分解する:

36 = 6^{2}

= 6^{2}

累乗根の規則を適用する:

n a^{n} = a

6^{2} = 6

= 6

v_{1, 2} = \frac{- ( - 2 ) \pm 6}{2 \cdot 1}

解を分離する

v_{1} = \frac{- ( - 2 ) + 6}{2 \cdot 1}, v_{2} = \frac{- ( - 2 ) - 6}{2 \cdot 1}

v = \frac{- ( - 2 ) + 6}{2 \cdot 1} : 4

\frac{- ( - 2 ) + 6}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{2 + 6}{2 \cdot 1}

数を足す:

2 + 6 = 8

= \frac{8}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{8}{2}

数を割る:

\frac{8}{2} = 4

= 4

v = \frac{- ( - 2 ) - 6}{2 \cdot 1} : - 2

\frac{- ( - 2 ) - 6}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{2 - 6}{2 \cdot 1}

数を引く:

2 - 6 = - 4

= \frac{- 4}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 4}{2}

分数の規則を適用する:

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

= - \frac{4}{2}

数を割る:

\frac{4}{2} = 2

= - 2

二次equationの解:

v = 4, v = - 2

v = 4, v = - 2

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

x^{2} = f (a)

の場合, 解は

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

u = 4, u = - 4

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

解く

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

u^{2} = - 2

x^{2} = f (a)

の場合, 解は

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

u = - 2, u = - - 2

簡素化

- 2 : 2 i

- 2

累乗根の規則を適用する:

- a = - 1 a

- 2 = - 1 2

= - 1 2

虚数の規則を適用する:

- 1 = i

= 2 i

簡素化

- - 2 : - 2 i

- - 2

簡素化

- 2 : 2 i

- 2

累乗根の規則を適用する:

- a = - 1 a

- 2 = - 1 2

= - 1 2

虚数の規則を適用する:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

解答は

u = 2, u = - 2, u = 2 i, u = - 2 i

代用を戻す

u = csc (x)

csc (x) = 2, csc (x) = - 2, csc (x) = 2 i, csc (x) = - 2 i

csc (x) = 2, csc (x) = - 2, csc (x) = 2 i, csc (x) = - 2 i

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

以下の一般解

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = - 2

以下の一般解

csc (x) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = 2 i :

解なし

csc (x) = 2 i

解なし

csc (x) = - 2 i :

解なし

csc (x) = - 2 i

解なし

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

グラフ

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