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人気のある三角関数 >

cot(x)cos(x)-sin(x)=1

解

cot (x) cos (x) - sin (x) = 1

解

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = 27 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

解答ステップ

cot (x) cos (x) - sin (x) = 1

両辺から

1

を引く

cot (x) cos (x) - sin (x) - 1 = 0

三角関数の公式を使用して書き換える

- 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 )}

= - 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 )}

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

分数を組み合わせる

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

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

元を分数に変換する:

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

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

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

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

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

sin (x) sin (x)

指数の規則を適用する:

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

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

= sin^{1 + 1} (x)

数を足す:

1 + 1 = 2

= sin^{2} (x)

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

類似した元を足す:

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

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

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

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

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

置換で解く

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

仮定:

sin (x) = u

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

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

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

以下で両辺を乗じる:

u

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

以下で両辺を乗じる:

u

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

簡素化

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

簡素化

- 1 \cdot u : - u

- 1 \cdot u

乗算:

1 \cdot u = u

= - u

簡素化

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

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

分数を乗じる:

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

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

共通因数を約分する:

u

= 1 - 2 u^{2}

簡素化

0 \cdot u : 0

0 \cdot u

規則を適用

0 \cdot a = 0

= 0

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

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

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

解く

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

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

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

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

規則を適用

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

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

= 1^{2}

規則を適用

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

数を乗じる:

4 \cdot 2 \cdot 1 = 8

= 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:

n a^{n} = a

3^{2} = 3

= 3

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

解を分離する

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

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

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

括弧を削除する:

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

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

数を足す:

1 + 3 = 4

= \frac{4}{- 2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{4}{- 4}

分数の規則を適用する:

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

= - \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= - 1

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

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

括弧を削除する:

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

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

数を引く:

1 - 3 = - 2

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

数を乗じる:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

分数の規則を適用する:

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

= \frac{2}{4}

共通因数を約分する:

2

= \frac{1}{2}

二次equationの解:

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

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

解を検算する

未定義の (特異) 点を求める:

u = 0

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

の分母をゼロに比較する

u = 0

以下の点は定義されていない

u = 0

未定義のポイントを解に組み合わせる:

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

代用を戻す

u = sin (x)

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

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

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

sin (x) = - 1

以下の一般解

sin (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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

以下の一般解

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

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}

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

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

すべての解を組み合わせる

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

人気の例

cos^4(x)+cos^3(x)-2=0

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

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

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

(sin^2(x)-2cos(x)+1)/4 =0

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

cos^2(x)+cos^4(x)+cos^6(x)=0

cos^{2} (x) + cos^{4} (x) + cos^{6} (x) = 0

sin (x) = \frac{- 1}{4}