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人気のある三角関数 >

csc(x)+cot(x)=(sqrt(3))/3

解

csc (x) + cot (x) = \frac{3}{3}

解

x = \frac{2 π}{3} + 2 πn

+1

度

x = 12 0^{\circ} + 36 0^{\circ} n

解答ステップ

csc (x) + cot (x) = \frac{3}{3}

両辺から

\frac{3}{3}

を引く

csc (x) + cot (x) - \frac{1}{3} = 0

簡素化

csc (x) + cot (x) - \frac{1}{3} : \frac{3 csc ( x ) + 3 cot ( x ) - 1}{3}

csc (x) + cot (x) - \frac{1}{3}

元を分数に変換する:

csc (x) = \frac{csc ( x ) 3}{3}, cot (x) = \frac{cot ( x ) 3}{3}

= \frac{csc ( x ) 3}{3} + \frac{cot ( x ) 3}{3} - \frac{1}{3}

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{csc ( x ) 3 + cot ( x ) 3 - 1}{3}

\frac{3 csc ( x ) + 3 cot ( x ) - 1}{3} = 0

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

3 csc (x) + 3 cot (x) - 1 = 0

サイン, コサインで表わす

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

簡素化

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

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

3 \frac{1}{sin ( x )} = \frac{3}{sin ( x )}

3 \frac{1}{sin ( x )}

分数を乗じる:

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

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

乗算:

1 \cdot 3 = 3

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

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

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

分数を乗じる:

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

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

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

分数を組み合わせる

\frac{3}{sin ( x )} + \frac{3 cos ( x )}{sin ( x )} : \frac{3 + 3 cos ( x )}{sin ( x )}

規則を適用

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

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

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

元を分数に変換する:

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

= \frac{3 + cos ( x ) 3}{sin ( x )} - \frac{1 \cdot 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{3 + cos ( x ) 3 - 1 \cdot sin ( x )}{sin ( x )}

乗算:

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

= \frac{3 + 3 cos ( x ) - sin ( x )}{sin ( x )}

\frac{3 + 3 cos ( x ) - sin ( x )}{sin ( x )} = 0

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

3 + 3 cos (x) - sin (x) = 0

両辺に

sin (x)

を足す

3 + 3 cos (x) = sin (x)

両辺を2乗する

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

両辺から

sin^{2} (x)

を引く

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

三角関数の公式を使用して書き換える

(3 + cos (x) 3)^{2} - 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)

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

簡素化

(3 + cos (x) 3)^{2} - (1 - cos^{2} (x)) : 4 cos^{2} (x) + 6 cos (x) + 2

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

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

(3 + cos (x) 3)^{2} : 3 + 6 cos (x) + 3 cos^{2} (x)

完全平方式を適用する:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 3, b = cos (x) 3

= (3)^{2} + 23 cos (x) 3 + (cos (x) 3)^{2}

簡素化

(3)^{2} + 23 cos (x) 3 + (cos (x) 3)^{2} : 3 + 6 cos (x) + 3 cos^{2} (x)

(3)^{2} + 23 cos (x) 3 + (cos (x) 3)^{2}

(3)^{2} = 3

(3)^{2}

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

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

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 3

23 cos (x) 3 = 6 cos (x)

23 cos (x) 3

累乗根の規則を適用する:

a a = a

33 = 3

= 2 \cdot 3 cos (x)

数を乗じる:

2 \cdot 3 = 6

= 6 cos (x)

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

(cos (x) 3)^{2}

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

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

(3)^{2} : 3

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

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

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 3

= cos^{2} (x) \cdot 3

= 3 + 6 cos (x) + 3 cos^{2} (x)

= 3 + 6 cos (x) + 3 cos^{2} (x)

= 3 + 6 cos (x) + 3 cos^{2} (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)

= 3 + 6 cos (x) + 3 cos^{2} (x) - 1 + cos^{2} (x)

簡素化

3 + 6 cos (x) + 3 cos^{2} (x) - 1 + cos^{2} (x) : 4 cos^{2} (x) + 6 cos (x) + 2

3 + 6 cos (x) + 3 cos^{2} (x) - 1 + cos^{2} (x)

条件のようなグループ

= 6 cos (x) + 3 cos^{2} (x) + cos^{2} (x) + 3 - 1

類似した元を足す:

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

= 6 cos (x) + 4 cos^{2} (x) + 3 - 1

数を足す/引く:

3 - 1 = 2

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

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

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

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

置換で解く

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

仮定:

cos (x) = u

2 + 4 u^{2} + 6 u = 0

2 + 4 u^{2} + 6 u = 0 : u = - \frac{1}{2}, u = - 1

2 + 4 u^{2} + 6 u = 0

標準的な形式で書く

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

4 u^{2} + 6 u + 2 = 0

解くとthe二次式

4 u^{2} + 6 u + 2 = 0

二次Equationの公式:

次の場合:

a = 4, b = 6, c = 2

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

6^{2} - 4 \cdot 4 \cdot 2 = 2

6^{2} - 4 \cdot 4 \cdot 2

数を乗じる:

4 \cdot 4 \cdot 2 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

数を引く:

36 - 32 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 6 \pm 2}{2 \cdot 4}

解を分離する

u_{1} = \frac{- 6 + 2}{2 \cdot 4}, u_{2} = \frac{- 6 - 2}{2 \cdot 4}

u = \frac{- 6 + 2}{2 \cdot 4} : - \frac{1}{2}

\frac{- 6 + 2}{2 \cdot 4}

数を足す/引く:

- 6 + 2 = - 4

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

数を乗じる:

2 \cdot 4 = 8

= \frac{- 4}{8}

分数の規則を適用する:

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

= - \frac{4}{8}

共通因数を約分する:

4

= - \frac{1}{2}

u = \frac{- 6 - 2}{2 \cdot 4} : - 1

\frac{- 6 - 2}{2 \cdot 4}

数を引く:

- 6 - 2 = - 8

= \frac{- 8}{2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{- 8}{8}

分数の規則を適用する:

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

= - \frac{8}{8}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

以下の一般解

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

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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 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 = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = π + 2 πn

元のequationに当てはめて解を検算する

csc (x) + cot (x) = \frac{3}{3}

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{2 π}{3} + 2 πn :

真

\frac{2 π}{3} + 2 πn

挿入

n = 1

\frac{2 π}{3} + 2 π 1

csc (x) + cot (x) = \frac{3}{3}

の挿入向け

x = \frac{2 π}{3} + 2 π 1

csc (\frac{2 π}{3} + 2 π 1) + cot (\frac{2 π}{3} + 2 π 1) = \frac{3}{3}

改良

0.57735 \dots = 0.57735 \dots

\Rightarrow 真

解答を確認する

\frac{4 π}{3} + 2 πn :

偽

\frac{4 π}{3} + 2 πn

挿入

n = 1

\frac{4 π}{3} + 2 π 1

csc (x) + cot (x) = \frac{3}{3}

の挿入向け

x = \frac{4 π}{3} + 2 π 1

csc (\frac{4 π}{3} + 2 π 1) + cot (\frac{4 π}{3} + 2 π 1) = \frac{3}{3}

改良

- 0.57735 \dots = 0.57735 \dots

\Rightarrow 偽

解答を確認する

π + 2 πn :

偽

π + 2 πn

挿入

n = 1

π + 2 π 1

csc (x) + cot (x) = \frac{3}{3}

の挿入向け

x = π + 2 π 1

csc (π + 2 π 1) + cot (π + 2 π 1) = \frac{3}{3}

未定義

\Rightarrow 偽

x = \frac{2 π}{3} + 2 πn

グラフ

人気の例

sin(θ)+1=cos(θ)

sin (θ) + 1 = cos (θ)

sqrt(2)sin^2(θ)-sin(θ)=0

2 sin^{2} (θ) - sin (θ) = 0

2sin^2(x)=2-sqrt(3)cos(x)

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

tan^2(x)= 3/2 sec(x)

tan^{2} (x) = \frac{3}{2} sec (x)

cot(θ)=cot^2(θ)

cot (θ) = cot^{2} (θ)