ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある三角関数 >

2cot^2(x)+2csc^2(x)=1+4csc(x)

解

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

解

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度

x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

1 + 4 csc (x)

を引く

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

三角関数の公式を使用して書き換える

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

簡素化

- 1 + 2 (csc^{2} (x) - 1) + 2 csc^{2} (x) - 4 csc (x) : 4 csc^{2} (x) - 4 csc (x) - 3

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

数を乗じる:

2 \cdot 1 = 2

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

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

簡素化

- 1 + 2 csc^{2} (x) - 2 + 2 csc^{2} (x) - 4 csc (x) : 4 csc^{2} (x) - 4 csc (x) - 3

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

条件のようなグループ

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

類似した元を足す:

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

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

数を引く:

- 1 - 2 = - 3

= 4 csc^{2} (x) - 4 csc (x) - 3

= 4 csc^{2} (x) - 4 csc (x) - 3

= 4 csc^{2} (x) - 4 csc (x) - 3

- 3 - 4 csc (x) + 4 csc^{2} (x) = 0

置換で解く

- 3 - 4 csc (x) + 4 csc^{2} (x) = 0

仮定:

csc (x) = u

- 3 - 4 u + 4 u^{2} = 0

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

- 3 - 4 u + 4 u^{2} = 0

標準的な形式で書く

a x^{2} + b x + c = 0

4 u^{2} - 4 u - 3 = 0

解くとthe二次式

4 u^{2} - 4 u - 3 = 0

二次Equationの公式:

次の場合:

a = 4, b = - 4, c = - 3

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 4 ( - 3 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 4 ( - 3 )}{2 \cdot 4}

(- 4)^{2} - 4 \cdot 4 (- 3) = 8

(- 4)^{2} - 4 \cdot 4 (- 3)

規則を適用

- (- a) = a

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

指数の規則を適用する:

n

が偶数であれば

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

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

= 4^{2} + 4 \cdot 4 \cdot 3

数を乗じる:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数を足す:

16 + 48 = 64

= 64

数を因数に分解する:

64 = 8^{2}

= 8^{2}

累乗根の規則を適用する:

n a^{n} = a

8^{2} = 8

= 8

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

解を分離する

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 4}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 4}

u = \frac{- ( - 4 ) + 8}{2 \cdot 4} : \frac{3}{2}

\frac{- ( - 4 ) + 8}{2 \cdot 4}

規則を適用

- (- a) = a

= \frac{4 + 8}{2 \cdot 4}

数を足す:

4 + 8 = 12

= \frac{12}{2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{12}{8}

共通因数を約分する:

4

= \frac{3}{2}

u = \frac{- ( - 4 ) - 8}{2 \cdot 4} : - \frac{1}{2}

\frac{- ( - 4 ) - 8}{2 \cdot 4}

規則を適用

- (- a) = a

= \frac{4 - 8}{2 \cdot 4}

数を引く:

4 - 8 = - 4

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

数を乗じる:

2 \cdot 4 = 8

= \frac{- 4}{8}

分数の規則を適用する:

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

= - \frac{4}{8}

共通因数を約分する:

4

= - \frac{1}{2}

二次equationの解:

u = \frac{3}{2}, u = - \frac{1}{2}

代用を戻す

u = csc (x)

csc (x) = \frac{3}{2}, csc (x) = - \frac{1}{2}

csc (x) = \frac{3}{2}, csc (x) = - \frac{1}{2}

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

csc (x) = \frac{3}{2}

三角関数の逆数プロパティを適用する

csc (x) = \frac{3}{2}

以下の一般解

csc (x) = \frac{3}{2}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

csc (x) = - \frac{1}{2} :

解なし

csc (x) = - \frac{1}{2}

csc (x) \leq - 1 or csc (x) \geq 1

解なし

すべての解を組み合わせる

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

10進法形式で解を証明する

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

グラフ

人気の例

sin(2θ)=cos(θ),0<= θ<2pi

sin (2 θ) = cos (θ), 0 \leq θ < 2 π

2cos^2(θ)-10=-cos(θ)-9

2 cos^{2} (θ) - 10 = - cos (θ) - 9

1 = tan (π + c)

6193cos(x)+2880cos(2x)=0

6193 cos (x) + 2880 cos (2 x) = 0

sin (x) = \frac{6}{11}