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人気のある三角関数 >

7+csc(2x)=3cot^2(2x)

解

7 + csc (2 x) = 3 cot^{2} (2 x)

解

x = - \frac{0.64350 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

+1

度

x = - 18.43494 \dots^{\circ} + 18 0^{\circ} n, x = 108.43494 \dots^{\circ} + 18 0^{\circ} n, x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

解答ステップ

7 + csc (2 x) = 3 cot^{2} (2 x)

両辺から

3 cot^{2} (2 x)

を引く

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

三角関数の公式を使用して書き換える

7 + csc (2 x) - 3 cot^{2} (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

= 7 + csc (2 x) - 3 (csc^{2} (2 x) - 1)

簡素化

7 + csc (2 x) - 3 (csc^{2} (2 x) - 1) : csc (2 x) - 3 csc^{2} (2 x) + 10

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

a = - 3, b = csc^{2} (2 x), c = 1

= - 3 csc^{2} (2 x) - (- 3) \cdot 1

マイナス・プラスの規則を適用する

- (- a) = a

= - 3 csc^{2} (2 x) + 3 \cdot 1

数を乗じる:

3 \cdot 1 = 3

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

= 7 + csc (2 x) - 3 csc^{2} (2 x) + 3

簡素化

7 + csc (2 x) - 3 csc^{2} (2 x) + 3 : csc (2 x) - 3 csc^{2} (2 x) + 10

7 + csc (2 x) - 3 csc^{2} (2 x) + 3

条件のようなグループ

= csc (2 x) - 3 csc^{2} (2 x) + 7 + 3

数を足す:

7 + 3 = 10

= csc (2 x) - 3 csc^{2} (2 x) + 10

= csc (2 x) - 3 csc^{2} (2 x) + 10

= csc (2 x) - 3 csc^{2} (2 x) + 10

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

置換で解く

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

仮定:

csc (2 x) = u

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

10 + u - 3 u^{2} = 0 : u = - \frac{5}{3}, u = 2

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

標準的な形式で書く

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

- 3 u^{2} + u + 10 = 0

解くとthe二次式

- 3 u^{2} + u + 10 = 0

二次Equationの公式:

次の場合:

a = - 3, b = 1, c = 10

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 3 ) \cdot 10}{2 ( - 3 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 3 ) \cdot 10}{2 ( - 3 )}

1^{2} - 4 (- 3) \cdot 10 = 11

1^{2} - 4 (- 3) \cdot 10

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 3) \cdot 10

規則を適用

- (- a) = a

= 1 + 4 \cdot 3 \cdot 10

数を乗じる:

4 \cdot 3 \cdot 10 = 120

= 1 + 120

数を足す:

1 + 120 = 121

= 121

数を因数に分解する:

121 = 1 1^{2}

= 1 1^{2}

累乗根の規則を適用する:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- 1 \pm 11}{2 ( - 3 )}

解を分離する

u_{1} = \frac{- 1 + 11}{2 ( - 3 )}, u_{2} = \frac{- 1 - 11}{2 ( - 3 )}

u = \frac{- 1 + 11}{2 ( - 3 )} : - \frac{5}{3}

\frac{- 1 + 11}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 + 11}{- 2 \cdot 3}

数を足す/引く:

- 1 + 11 = 10

= \frac{10}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{10}{- 6}

分数の規則を適用する:

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

= - \frac{10}{6}

共通因数を約分する:

2

= - \frac{5}{3}

u = \frac{- 1 - 11}{2 ( - 3 )} : 2

\frac{- 1 - 11}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 - 11}{- 2 \cdot 3}

数を引く:

- 1 - 11 = - 12

= \frac{- 12}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

分数の規則を適用する:

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

= \frac{12}{6}

数を割る:

\frac{12}{6} = 2

= 2

二次equationの解:

u = - \frac{5}{3}, u = 2

代用を戻す

u = csc (2 x)

csc (2 x) = - \frac{5}{3}, csc (2 x) = 2

csc (2 x) = - \frac{5}{3}, csc (2 x) = 2

csc (2 x) = - \frac{5}{3} : x = - \frac{arccsc ( \frac{5}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + πn

csc (2 x) = - \frac{5}{3}

三角関数の逆数プロパティを適用する

csc (2 x) = - \frac{5}{3}

以下の一般解

csc (2 x) = - \frac{5}{3}

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

2 x = arccsc (- \frac{5}{3}) + 2 πn, 2 x = π + arccsc (\frac{5}{3}) + 2 πn

2 x = arccsc (- \frac{5}{3}) + 2 πn, 2 x = π + arccsc (\frac{5}{3}) + 2 πn

解く

2 x = arccsc (- \frac{5}{3}) + 2 πn : x = - \frac{arccsc ( \frac{5}{3} )}{2} + πn

2 x = arccsc (- \frac{5}{3}) + 2 πn

簡素化

arccsc (- \frac{5}{3}) + 2 πn : - arccsc (\frac{5}{3}) + 2 πn

arccsc (- \frac{5}{3}) + 2 πn

次のプロパティを使用する:

arccsc (- x) = - arccsc (x)

arccsc (- \frac{5}{3}) = - arccsc (\frac{5}{3})

= - arccsc (\frac{5}{3}) + 2 πn

2 x = - arccsc (\frac{5}{3}) + 2 πn

以下で両辺を割る

2

2 x = - arccsc (\frac{5}{3}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = - \frac{arccsc ( \frac{5}{3} )}{2} + \frac{2 πn}{2}

簡素化

x = - \frac{arccsc ( \frac{5}{3} )}{2} + πn

x = - \frac{arccsc ( \frac{5}{3} )}{2} + πn

解く

2 x = π + arccsc (\frac{5}{3}) + 2 πn : x = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + πn

2 x = π + arccsc (\frac{5}{3}) + 2 πn

以下で両辺を割る

2

2 x = π + arccsc (\frac{5}{3}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + \frac{2 πn}{2}

簡素化

x = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + πn

x = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + πn

x = - \frac{arccsc ( \frac{5}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + πn

csc (2 x) = 2 : x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

csc (2 x) = 2

以下の一般解

csc (2 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

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

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

解く

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{π}{6} + 2 πn

以下で両辺を割る

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

解く

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{5 π}{6} + 2 πn

以下で両辺を割る

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

すべての解を組み合わせる

x = - \frac{arccsc ( \frac{5}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arccsc ( \frac{5}{3} )}{2} + πn, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

10進法形式で解を証明する

x = - \frac{0.64350 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

グラフ

人気の例

cos(θ)=cos(pi/(13))

cos (θ) = cos (\frac{π}{13})

sin(2x+pi/6)=-1

sin (2 x + \frac{π}{6}) = - 1

tan (x) = 2 π

2sin^2(x)-sqrt(2)=0

2 sin^{2} (x) - 2 = 0

solvefor x,z=sin(5x)y^3

so l v e f or x, z = sin (5 x) y^{3}