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人気のある三角関数 >

csc^4(x)-1=5cot^2(x)

解

csc^{4} (x) - 1 = 5 cot^{2} (x)

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, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 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, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

解答ステップ

csc^{4} (x) - 1 = 5 cot^{2} (x)

両辺から

5 cot^{2} (x)

を引く

csc^{4} (x) - 1 - 5 cot^{2} (x) = 0

三角関数の公式を使用して書き換える

- 1 + csc^{4} (x) - 5 cot^{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

= - 1 + csc^{4} (x) - 5 (csc^{2} (x) - 1)

簡素化

- 1 + csc^{4} (x) - 5 (csc^{2} (x) - 1) : csc^{4} (x) - 5 csc^{2} (x) + 4

- 1 + csc^{4} (x) - 5 (csc^{2} (x) - 1)

拡張

- 5 (csc^{2} (x) - 1) : - 5 csc^{2} (x) + 5

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

分配法則を適用する:

a (b - c) = ab - a c

a = - 5, b = csc^{2} (x), c = 1

= - 5 csc^{2} (x) - (- 5) \cdot 1

マイナス・プラスの規則を適用する

- (- a) = a

= - 5 csc^{2} (x) + 5 \cdot 1

数を乗じる:

5 \cdot 1 = 5

= - 5 csc^{2} (x) + 5

= - 1 + csc^{4} (x) - 5 csc^{2} (x) + 5

簡素化

- 1 + csc^{4} (x) - 5 csc^{2} (x) + 5 : csc^{4} (x) - 5 csc^{2} (x) + 4

- 1 + csc^{4} (x) - 5 csc^{2} (x) + 5

条件のようなグループ

= csc^{4} (x) - 5 csc^{2} (x) - 1 + 5

数を足す/引く:

- 1 + 5 = 4

= csc^{4} (x) - 5 csc^{2} (x) + 4

= csc^{4} (x) - 5 csc^{2} (x) + 4

= csc^{4} (x) - 5 csc^{2} (x) + 4

4 + csc^{4} (x) - 5 csc^{2} (x) = 0

置換で解く

4 + csc^{4} (x) - 5 csc^{2} (x) = 0

仮定:

csc (x) = u

4 + u^{4} - 5 u^{2} = 0

4 + u^{4} - 5 u^{2} = 0 : u = 2, u = - 2, u = 1, u = - 1

4 + u^{4} - 5 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} - 5 u^{2} + 4 = 0

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

v^{2} - 5 v + 4 = 0

解く

v^{2} - 5 v + 4 = 0 : v = 4, v = 1

v^{2} - 5 v + 4 = 0

解くとthe二次式

v^{2} - 5 v + 4 = 0

二次Equationの公式:

次の場合:

a = 1, b = - 5, c = 4

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot 4 = 3

(- 5)^{2} - 4 \cdot 1 \cdot 4

指数の規則を適用する:

n

が偶数であれば

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

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 4

数を乗じる:

4 \cdot 1 \cdot 4 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

数を引く:

25 - 16 = 9

= 9

数を因数に分解する:

9 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 3

v_{1, 2} = \frac{- ( - 5 ) \pm 3}{2 \cdot 1}

解を分離する

v_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 1}, v_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 1}

v = \frac{- ( - 5 ) + 3}{2 \cdot 1} : 4

\frac{- ( - 5 ) + 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 + 3}{2 \cdot 1}

数を足す:

5 + 3 = 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{- ( - 5 ) - 3}{2 \cdot 1} : 1

\frac{- ( - 5 ) - 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 - 3}{2 \cdot 1}

数を引く:

5 - 3 = 2

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

数を乗じる:

2 \cdot 1 = 2

= \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

v = 4, v = 1

v = 4, v = 1

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} = 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

解答は

u = 2, u = - 2, u = 1, u = - 1

代用を戻す

u = csc (x)

csc (x) = 2, csc (x) = - 2, csc (x) = 1, csc (x) = - 1

csc (x) = 2, csc (x) = - 2, csc (x) = 1, csc (x) = - 1

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) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

以下の一般解

csc (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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

csc (x) = - 1

以下の一般解

csc (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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

グラフ

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