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人気のある三角関数 >

((1+cot^2(x)))/(cos^2(x))=cot^2(x)

解

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

解

以下の解はない : x \in R

解答ステップ

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

両辺から

cot^{2} (x)

を引く

\frac{1 + cot ^{2} ( x )}{cos ^{2} ( x )} - cot^{2} (x) = 0

簡素化

\frac{1 + cot ^{2} ( x )}{cos ^{2} ( x )} - cot^{2} (x) : \frac{1 + cot ^{2} ( x ) - cot ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{1 + cot ^{2} ( x )}{cos ^{2} ( x )} - cot^{2} (x)

元を分数に変換する:

cot^{2} (x) = \frac{cot ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{1 + cot ^{2} ( x )}{cos ^{2} ( x )} - \frac{cot ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( 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{1 + cot ^{2} ( x ) - cot ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{1 + cot ^{2} ( x ) - cot ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

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

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

三角関数の公式を使用して書き換える

1 + cot^{2} (x) - cos^{2} (x) cot^{2} (x)

ピタゴラスの公式を使用する:

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

= - cos^{2} (x) cot^{2} (x) + csc^{2} (x)

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

因数

csc^{2} (x) - cos^{2} (x) cot^{2} (x) : (csc (x) + cos (x) cot (x)) (csc (x) - cos (x) cot (x))

csc^{2} (x) - cos^{2} (x) cot^{2} (x)

cos^{2} (x) cot^{2} (x)

を書き換え

(cos (x) cot (x))^{2}

cos^{2} (x) cot^{2} (x)

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

cos^{2} (x) cot^{2} (x) = (cos (x) cot (x))^{2}

= (cos (x) cot (x))^{2}

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

2乗の差の公式を適用する:

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

csc^{2} (x) - (cos (x) cot (x))^{2} = (csc (x) + cos (x) cot (x)) (csc (x) - cos (x) cot (x))

= (csc (x) + cos (x) cot (x)) (csc (x) - cos (x) cot (x))

(csc (x) + cos (x) cot (x)) (csc (x) - cos (x) cot (x)) = 0

各部分を別個に解く

csc (x) + cos (x) cot (x) = 0 or csc (x) - cos (x) cot (x) = 0

csc (x) + cos (x) cot (x) = 0 :

解なし

csc (x) + cos (x) cot (x) = 0

サイン, コサインで表わす

csc (x) + cos (x) cot (x)

基本的な三角関数の公式を使用する:

csc (x) = \frac{1}{sin ( x )}

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

基本的な三角関数の公式を使用する:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

簡素化

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

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

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

cos (x) \frac{cos ( x )}{sin ( x )}

分数を乗じる:

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

= \frac{cos ( x ) cos ( x )}{sin ( x )}

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

cos (x) cos (x)

指数の規則を適用する:

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

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

= cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= cos^{2} (x)

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

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

規則を適用

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

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

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

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

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

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

置換で解く

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

仮定:

cos (x) = u

1 + u^{2} = 0

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

1 + u^{2} = 0

1

を右側に移動します

1 + u^{2} = 0

両辺から

1

を引く

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

簡素化

u^{2} = - 1

u^{2} = - 1

x^{2} = f (a)

の場合, 解は

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

u = - 1, u = - - 1

簡素化

- 1 : i

- 1

虚数の規則を適用する:

- 1 = i

= i

簡素化

- - 1 : - i

- - 1

虚数の規則を適用する:

- 1 = i

= - i

u = i, u = - i

代用を戻す

u = cos (x)

cos (x) = i, cos (x) = - i

cos (x) = i, cos (x) = - i

cos (x) = i :

解なし

cos (x) = i

解なし

cos (x) = - i :

解なし

cos (x) = - i

解なし

すべての解を組み合わせる

解なし

csc (x) - cos (x) cot (x) = 0 :

解なし

csc (x) - cos (x) cot (x) = 0

サイン, コサインで表わす

csc (x) - cos (x) cot (x)

基本的な三角関数の公式を使用する:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} - cos (x) cot (x)

基本的な三角関数の公式を使用する:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{1}{sin ( x )} - cos (x) \frac{cos ( x )}{sin ( x )}

簡素化

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

\frac{1}{sin ( x )} - cos (x) \frac{cos ( x )}{sin ( x )}

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

cos (x) \frac{cos ( x )}{sin ( x )}

分数を乗じる:

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

= \frac{cos ( x ) cos ( x )}{sin ( x )}

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

cos (x) cos (x)

指数の規則を適用する:

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

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

= cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= cos^{2} (x)

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

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

規則を適用

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

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

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

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

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

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

置換で解く

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

仮定:

cos (x) = u

1 - u^{2} = 0

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

1 - u^{2} = 0

1

を右側に移動します

1 - u^{2} = 0

両辺から

1

を引く

1 - u^{2} - 1 = 0 - 1

簡素化

- u^{2} = - 1

- u^{2} = - 1

以下で両辺を割る

- 1

- u^{2} = - 1

以下で両辺を割る

- 1

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

簡素化

u^{2} = 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

代用を戻す

u = cos (x)

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

以下の一般解

cos (x) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

以下の一般解

cos (x) = - 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn

equationは以下で未定義のため:

2 πn, π + 2 πn

解なし

すべての解を組み合わせる

以下の解はない : x \in R

グラフ

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