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人気のある >

(tan^2(3x))(csc(3x))+1=-1*sec^2(3x)

解

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

解

以下の解はない : x \in R

解答ステップ

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

両辺から

- 1 sec^{2} (3 x)

を引く

tan^{2} (3 x) csc (3 x) + 1 + sec^{2} (3 x) = 0

サイン, コサインで表わす

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )} + 1 + (\frac{1}{cos ( 3 x )})^{2} = 0

簡素化

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )} + 1 + (\frac{1}{cos ( 3 x )})^{2} : \frac{sin ( 3 x ) + 1 + cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )} + 1 + (\frac{1}{cos ( 3 x )})^{2}

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )} = \frac{sin ( 3 x )}{cos ^{2} ( 3 x )}

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )}

分数を乗じる:

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

= \frac{1 \cdot ( \frac{s i n ( 3 x )}{c o s ( 3 x )} ) ^{2}}{sin ( 3 x )}

乗算:

1 \cdot (\frac{sin ( 3 x )}{cos ( 3 x )})^{2} = (\frac{sin ( 3 x )}{cos ( 3 x )})^{2}

= \frac{( \frac{s i n ( 3 x )}{c o s ( 3 x )} ) ^{2}}{sin ( 3 x )}

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} = \frac{sin ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2}

指数の規則を適用する:

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

= \frac{sin ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

= \frac{\frac{s i n ^{2} ( 3 x )}{c o s ^{2} ( 3 x )}}{sin ( 3 x )}

分数の規則を適用する:

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

= \frac{sin ^{2} ( 3 x )}{cos ^{2} ( 3 x ) sin ( 3 x )}

共通因数を約分する:

sin (3 x)

= \frac{sin ( 3 x )}{cos ^{2} ( 3 x )}

(\frac{1}{cos ( 3 x )})^{2} = \frac{1}{cos ^{2} ( 3 x )}

(\frac{1}{cos ( 3 x )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{cos ^{2} ( 3 x )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( 3 x )}

= \frac{sin ( 3 x )}{cos ^{2} ( 3 x )} + 1 + \frac{1}{cos ^{2} ( 3 x )}

分数を組み合わせる

\frac{sin ( 3 x )}{cos ^{2} ( 3 x )} + \frac{1}{cos ^{2} ( 3 x )} : \frac{sin ( 3 x ) + 1}{cos ^{2} ( 3 x )}

規則を適用

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

= \frac{sin ( 3 x ) + 1}{cos ^{2} ( 3 x )}

= \frac{sin ( 3 x ) + 1}{cos ^{2} ( 3 x )} + 1

元を分数に変換する:

1 = \frac{1 cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

= \frac{sin ( 3 x ) + 1}{cos ^{2} ( 3 x )} + \frac{1 \cdot cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

分母が等しいので, 分数を組み合わせる:

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

= \frac{sin ( 3 x ) + 1 + 1 \cdot cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

乗算:

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

= \frac{sin ( 3 x ) + 1 + cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

\frac{sin ( 3 x ) + 1 + cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (3 x) + 1 + cos^{2} (3 x) = 0

両辺から

cos^{2} (3 x)

を引く

sin (3 x) + 1 = - cos^{2} (3 x)

両辺を2乗する

(sin (3 x) + 1)^{2} = (- cos^{2} (3 x))^{2}

両辺から

(- cos^{2} (3 x))^{2}

を引く

(sin (3 x) + 1)^{2} - cos^{4} (3 x) = 0

因数

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

(sin (3 x) + 1)^{2} - cos^{4} (3 x)

指数の規則を適用する:

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

cos^{4} (3 x) = (cos^{2} (3 x))^{2}

= (sin (3 x) + 1)^{2} - (cos^{2} (3 x))^{2}

2乗の差の公式を適用する:

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

(sin (3 x) + 1)^{2} - (cos^{2} (3 x))^{2} = ((sin (3 x) + 1) + cos^{2} (3 x)) ((sin (3 x) + 1) - cos^{2} (3 x))

= ((sin (3 x) + 1) + cos^{2} (3 x)) ((sin (3 x) + 1) - cos^{2} (3 x))

改良

= (cos^{2} (3 x) + sin (3 x) + 1) (sin (3 x) - cos^{2} (3 x) + 1)

(sin (3 x) + 1 + cos^{2} (3 x)) (sin (3 x) + 1 - cos^{2} (3 x)) = 0

各部分を別個に解く

sin (3 x) + 1 + cos^{2} (3 x) = 0 or sin (3 x) + 1 - cos^{2} (3 x) = 0

sin (3 x) + 1 + cos^{2} (3 x) = 0 : x = \frac{π}{2} + \frac{2 πn}{3}

sin (3 x) + 1 + cos^{2} (3 x) = 0

三角関数の公式を使用して書き換える

1 + cos^{2} (3 x) + sin (3 x)

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

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

簡素化

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

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

置換で解く

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

仮定:

sin (3 x) = u

2 + u - u^{2} = 0

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

2 + u - u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

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

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 2

規則を適用

- (- a) = a

= 1 + 4 \cdot 1 \cdot 2

数を乗じる:

4 \cdot 1 \cdot 2 = 8

= 1 + 8

数を足す:

1 + 8 = 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:

3^{2} = 3

= 3

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

解を分離する

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

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

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

括弧を削除する:

(- a) = - a

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

数を足す/引く:

- 1 + 3 = 2

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

数を乗じる:

2 \cdot 1 = 2

= \frac{2}{- 2}

分数の規則を適用する:

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

= - \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= - 1

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

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

括弧を削除する:

(- a) = - a

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

数を引く:

- 1 - 3 = - 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:

u = - 1, u = 2

代用を戻す

u = sin (3 x)

sin (3 x) = - 1, sin (3 x) = 2

sin (3 x) = - 1, sin (3 x) = 2

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

sin (3 x) = - 1

以下の一般解

sin (3 x) = - 1

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}

3 x = \frac{3 π}{2} + 2 πn

3 x = \frac{3 π}{2} + 2 πn

解く

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

3 x = \frac{3 π}{2} + 2 πn

以下で両辺を割る

3

3 x = \frac{3 π}{2} + 2 πn

以下で両辺を割る

3

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

簡素化

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

簡素化

\frac{3 x}{3} : x

\frac{3 x}{3}

数を割る:

\frac{3}{3} = 1

= x

簡素化

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3} : \frac{π}{2} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

分数の規則を適用する:

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

= \frac{3 π}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{3 π}{6}

共通因数を約分する:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

sin (3 x) = 2 :

解なし

sin (3 x) = 2

- 1 \leq sin (x) \leq 1

解なし

すべての解を組み合わせる

x = \frac{π}{2} + \frac{2 πn}{3}

sin (3 x) + 1 - cos^{2} (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

sin (3 x) + 1 - cos^{2} (3 x) = 0

三角関数の公式を使用して書き換える

1 - cos^{2} (3 x) + sin (3 x)

ピタゴラスの公式を使用する:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= sin (3 x) + sin^{2} (3 x)

sin (3 x) + sin^{2} (3 x) = 0

置換で解く

sin (3 x) + sin^{2} (3 x) = 0

仮定:

sin (3 x) = u

u + u^{2} = 0

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

u + u^{2} = 0

標準的な形式で書く

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

u^{2} + u = 0

解くとthe二次式

u^{2} + u = 0

二次Equationの公式:

次の場合:

a = 1, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot 0 = 1

1^{2} - 4 \cdot 1 \cdot 0

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 1 - 0

数を引く:

1 - 0 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 \cdot 1}

解を分離する

u_{1} = \frac{- 1 + 1}{2 \cdot 1}, u_{2} = \frac{- 1 - 1}{2 \cdot 1}

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

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

数を足す/引く:

- 1 + 1 = 0

= \frac{0}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{0}{2}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= 0

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

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

数を引く:

- 1 - 1 = - 2

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

数を乗じる:

2 \cdot 1 = 2

= \frac{- 2}{2}

分数の規則を適用する:

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

= - \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

u = 0, u = - 1

代用を戻す

u = sin (3 x)

sin (3 x) = 0, sin (3 x) = - 1

sin (3 x) = 0, sin (3 x) = - 1

sin (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin (3 x) = 0

以下の一般解

sin (3 x) = 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}

3 x = 0 + 2 πn, 3 x = π + 2 πn

3 x = 0 + 2 πn, 3 x = π + 2 πn

解く

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

以下で両辺を割る

3

3 x = 2 πn

以下で両辺を割る

3

\frac{3 x}{3} = \frac{2 πn}{3}

簡素化

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

解く

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

3 x = π + 2 πn

以下で両辺を割る

3

3 x = π + 2 πn

以下で両辺を割る

3

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

簡素化

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

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

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

sin (3 x) = - 1

以下の一般解

sin (3 x) = - 1

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}

3 x = \frac{3 π}{2} + 2 πn

3 x = \frac{3 π}{2} + 2 πn

解く

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

3 x = \frac{3 π}{2} + 2 πn

以下で両辺を割る

3

3 x = \frac{3 π}{2} + 2 πn

以下で両辺を割る

3

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

簡素化

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

簡素化

\frac{3 x}{3} : x

\frac{3 x}{3}

数を割る:

\frac{3}{3} = 1

= x

簡素化

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3} : \frac{π}{2} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

分数の規則を適用する:

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

= \frac{3 π}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{3 π}{6}

共通因数を約分する:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

すべての解を組み合わせる

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

すべての解を組み合わせる

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

元のequationに当てはめて解を検算する

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

に当てはめて解を確認する

equationに一致しないものを削除する。

解答を確認する

\frac{π}{2} + \frac{2 πn}{3} :

真

\frac{π}{2} + \frac{2 πn}{3}

挿入

n = 1

\frac{π}{2} + \frac{2 π 1}{3}

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

の挿入向け

x = \frac{π}{2} + \frac{2 π 1}{3}

tan^{2} (3 (\frac{π}{2} + \frac{2 π 1}{3})) csc (3 (\frac{π}{2} + \frac{2 π 1}{3})) + 1 = - 1 \cdot sec^{2} (3 (\frac{π}{2} + \frac{2 π 1}{3}))

改良

- \infty = - \infty

\Rightarrow 真

解答を確認する

\frac{2 πn}{3} :

偽

\frac{2 πn}{3}

挿入

n = 1

\frac{2 π 1}{3}

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

の挿入向け

x = \frac{2 π 1}{3}

tan^{2} (3 \cdot \frac{2 π 1}{3}) csc (3 \cdot \frac{2 π 1}{3}) + 1 = - 1 \cdot sec^{2} (3 \cdot \frac{2 π 1}{3})

未定義

\Rightarrow 偽

解答を確認する

\frac{π}{3} + \frac{2 πn}{3} :

偽

\frac{π}{3} + \frac{2 πn}{3}

挿入

n = 1

\frac{π}{3} + \frac{2 π 1}{3}

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

の挿入向け

x = \frac{π}{3} + \frac{2 π 1}{3}

tan^{2} (3 (\frac{π}{3} + \frac{2 π 1}{3})) csc (3 (\frac{π}{3} + \frac{2 π 1}{3})) + 1 = - 1 \cdot sec^{2} (3 (\frac{π}{3} + \frac{2 π 1}{3}))

未定義

\Rightarrow 偽

x = \frac{π}{2} + \frac{2 πn}{3}

equationは以下で未定義のため:

\frac{π}{2} + \frac{2 πn}{3}

以下の解はない : x \in R

人気の例

(4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 3

7=10sin(40(0-h))+17

7 = 10 sin (40 (0 - h)) + 17

sin(2x)=cos(x),0<= x<2pi

sin (2 x) = cos (x), 0 \leq x < 2 π

sec (x) = - \frac{5}{3}

(sin(x))/(1+cos(x))=sin(x/2)

\frac{sin ( x )}{1 + cos ( x )} = sin (\frac{x}{2})