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人気のある >

1+tan^2(x)cos(x)=sec(x)

解

1 + tan^{2} (x) cos (x) = sec (x)

解

x = 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n

解答ステップ

1 + tan^{2} (x) cos (x) = sec (x)

両辺から

sec (x)

を引く

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

サイン, コサインで表わす

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

簡素化

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

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

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

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

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

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

指数の規則を適用する:

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

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

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

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

分数を組み合わせる

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

規則を適用

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

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot cos (x) = cos (x)

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

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

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

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

両辺から

sin^{2} (x)

を引く

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

両辺を2乗する

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

両辺から

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

を引く

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

因数

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

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

指数の規則を適用する:

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

sin^{4} (x) = (sin^{2} (x))^{2}

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

2乗の差の公式を適用する:

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

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

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

改良

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

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

各部分を別個に解く

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

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

置換で解く

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

仮定:

cos (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 ( - 1 ) \cdot 0}{2 ( - 1 )}

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

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

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 0

規則を適用

- (- a) = a

= 1 + 4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 1 + 0

数を足す:

1 + 0 = 1

= 1

規則を適用

1 = 1

= 1

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

解を分離する

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

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

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

括弧を削除する:

(- a) = - a

= \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}

分数の規則を適用する:

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

= - \frac{0}{2}

規則を適用

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

= - 0

= 0

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

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

括弧を削除する:

(- a) = - a

= \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 = cos (x)

cos (x) = 0, cos (x) = 1

cos (x) = 0, cos (x) = 1

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

cos (x) = 0

以下の一般解

cos (x) = 0

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

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

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

すべての解を組み合わせる

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

簡素化

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

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

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

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

括弧を分配する

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

マイナス・プラスの規則を適用する

- (- a) = a, - (a) = - a

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

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

簡素化

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

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

条件のようなグループ

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

数を引く:

- 1 - 1 = - 2

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

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

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

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

置換で解く

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

仮定:

cos (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 \cdot 1 \cdot ( - 2 )}{2 \cdot 1}

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

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

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 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 \cdot 1}

解を分離する

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

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

\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}

規則を適用

\frac{a}{a} = 1

= 1

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

\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 = cos (x)

cos (x) = 1, cos (x) = - 2

cos (x) = 1, cos (x) = - 2

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) = - 2 :

解なし

cos (x) = - 2

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

x = 2 πn

すべての解を組み合わせる

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

元のequationに当てはめて解を検算する

1 + tan^{2} (x) cos (x) = sec (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} + 2 πn :

偽

\frac{π}{2} + 2 πn

挿入

n = 1

\frac{π}{2} + 2 π 1

1 + tan^{2} (x) cos (x) = sec (x)

の挿入向け

x = \frac{π}{2} + 2 π 1

1 + tan^{2} (\frac{π}{2} + 2 π 1) cos (\frac{π}{2} + 2 π 1) = sec (\frac{π}{2} + 2 π 1)

未定義

\Rightarrow 偽

解答を確認する

\frac{3 π}{2} + 2 πn :

偽

\frac{3 π}{2} + 2 πn

挿入

n = 1

\frac{3 π}{2} + 2 π 1

1 + tan^{2} (x) cos (x) = sec (x)

の挿入向け

x = \frac{3 π}{2} + 2 π 1

1 + tan^{2} (\frac{3 π}{2} + 2 π 1) cos (\frac{3 π}{2} + 2 π 1) = sec (\frac{3 π}{2} + 2 π 1)

未定義

\Rightarrow 偽

解答を確認する

2 πn :

真

2 πn

挿入

n = 1

2 π 1

1 + tan^{2} (x) cos (x) = sec (x)

の挿入向け

x = 2 π 1

1 + tan^{2} (2 π 1) cos (2 π 1) = sec (2 π 1)

改良

1 = 1

\Rightarrow 真

x = 2 πn

人気の例

sec(θ)=(sqrt(6))/2

sec (θ) = \frac{6}{2}

sin^2(x)+cos^2(x)=sin(x)csc^2(x)

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

1.5sqrt(2)=3cos(x)

1.5 2 = 3 cos (x)

(sin(59))/8 =(sin(x))/(9.25)

\frac{sin ( 5 9 ^{\circ} )}{8} = \frac{sin ( x )}{9.25}

2 a \cdot cos (θ) = 0