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人気のある三角関数 >

cos(x)sec(x)cot^2(x)=csc^2(x)

解

cos (x) \cdot sec (x) \cdot cot^{2} (x) = csc^{2} (x)

解

以下の解はない : x \in R

解答ステップ

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

両辺から

csc^{2} (x)

を引く

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

サイン, コサインで表わす

- csc^{2} (x) + cos (x) cot^{2} (x) sec (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 )})^{2} + cos (x) cot^{2} (x) sec (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 )})^{2} + cos (x) (\frac{cos ( x )}{sin ( x )})^{2} sec (x)

基本的な三角関数の公式を使用する:

sec (x) = \frac{1}{cos ( x )}

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

簡素化

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

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

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

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

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

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

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

指数の規則を適用する:

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

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

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

乗算:

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

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

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

規則を適用

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

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

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

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

両辺に

1

を足す

- 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

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

以下の解はない : x \in R

グラフ

人気の例

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4 cot (x) = 3 csc (x)

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cos(x)=(sqrt(5))/3

cos (x) = \frac{5}{3}

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sin(x)=-cos(2x)

sin (x) = - cos (2 x)