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人気のある三角関数 >

cos(x)cot(x)-3(1-sin(x))=0

解

cos (x) cot (x) - 3 (1 - sin (x)) = 0

解

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

+1

度

x = 9 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

cos (x) cot (x) - 3 (1 - sin (x)) = 0

三角関数の公式を使用して書き換える

- (1 - sin (x)) \cdot 3 + 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)) \cdot 3 + 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 )}

= - 3 (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)

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

簡素化

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

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

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

拡張

- 3 (1 - sin (x)) : - 3 + 3 sin (x)

- 3 (1 - sin (x))

分配法則を適用する:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin (x)

= - 3 \cdot 1 - (- 3) sin (x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 3 \cdot 1 + 3 sin (x)

数を乗じる:

3 \cdot 1 = 3

= - 3 + 3 sin (x)

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

分数を組み合わせる

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

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

元を分数に変換する:

3 sin (x) = \frac{3 sin ( x ) sin ( x )}{sin ( x )}

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

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

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

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

3 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)

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

数を足す:

1 + 1 = 2

= 3 sin^{2} (x)

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

類似した元を足す:

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

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

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

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

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

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

置換で解く

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

仮定:

sin (x) = u

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

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

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

以下で両辺を乗じる:

u

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

以下で両辺を乗じる:

u

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

簡素化

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

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

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

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

解く

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

- 3 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} - 3 u + 1 = 0

解くとthe二次式

2 u^{2} - 3 u + 1 = 0

二次Equationの公式:

次の場合:

a = 2, b = - 3, c = 1

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 \cdot 1}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 \cdot 1}{2 \cdot 2}

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

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

指数の規則を適用する:

n

が偶数であれば

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

(- 3)^{2} = 3^{2}

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

数を乗じる:

4 \cdot 2 \cdot 1 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数を引く:

9 - 8 = 1

= 1

規則を適用

1 = 1

= 1

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

解を分離する

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

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

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

規則を適用

- (- a) = a

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

数を足す:

3 + 1 = 4

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

数を乗じる:

2 \cdot 2 = 4

= \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= 1

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

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

規則を適用

- (- a) = a

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

数を引く:

3 - 1 = 2

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

数を乗じる:

2 \cdot 2 = 4

= \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

- 3 + \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{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{π}{2} + 2 πn

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

グラフ

人気の例

18= 1/2*9*5*sin(θ)

18 = \frac{1}{2} \cdot 9 \cdot 5 \cdot sin (θ)

sqrt(2)cos^2(x)=cos^2(x)

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

sin (\frac{π x}{2}) = 0

sin(x)= 1/2 sqrt(2)

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

8cos(2x)=2sin(x)+7

8 cos (2 x) = 2 sin (x) + 7