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人気のある三角関数 >

csc(x)+cot(x)=3

解

csc (x) + cot (x) = 3

解

x = 0.64350 \dots + 2 πn

+1

度

x = 36.86989 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

csc (x) + cot (x) = 3

両辺から

3

を引く

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

サイン, コサインで表わす

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

簡素化

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

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

分数を組み合わせる

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

規則を適用

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

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

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

元を分数に変換する:

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

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

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

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

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

両辺に

3 sin (x)

を足す

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

両辺を2乗する

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

両辺から

(3 sin (x))^{2}

を引く

(1 + cos (x))^{2} - 9 sin^{2} (x) = 0

三角関数の公式を使用して書き換える

(1 + cos (x))^{2} - 9 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))^{2} - 9 (1 - cos^{2} (x))

簡素化

(1 + cos (x))^{2} - 9 (1 - cos^{2} (x)) : 10 cos^{2} (x) + 2 cos (x) - 8

(1 + cos (x))^{2} - 9 (1 - cos^{2} (x))

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

完全平方式を適用する:

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

a = 1, b = cos (x)

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

簡素化

1^{2} + 2 \cdot 1 \cdot cos (x) + cos^{2} (x) : 1 + 2 cos (x) + cos^{2} (x)

1^{2} + 2 \cdot 1 \cdot cos (x) + cos^{2} (x)

規則を適用

1^{a} = 1

1^{2} = 1

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

数を乗じる:

2 \cdot 1 = 2

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

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

= 1 + 2 cos (x) + cos^{2} (x) - 9 (1 - cos^{2} (x))

拡張

- 9 (1 - cos^{2} (x)) : - 9 + 9 cos^{2} (x)

- 9 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = - 9, b = 1, c = cos^{2} (x)

= - 9 \cdot 1 - (- 9) cos^{2} (x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 9 \cdot 1 + 9 cos^{2} (x)

数を乗じる:

9 \cdot 1 = 9

= - 9 + 9 cos^{2} (x)

= 1 + 2 cos (x) + cos^{2} (x) - 9 + 9 cos^{2} (x)

簡素化

1 + 2 cos (x) + cos^{2} (x) - 9 + 9 cos^{2} (x) : 10 cos^{2} (x) + 2 cos (x) - 8

1 + 2 cos (x) + cos^{2} (x) - 9 + 9 cos^{2} (x)

条件のようなグループ

= 2 cos (x) + cos^{2} (x) + 9 cos^{2} (x) + 1 - 9

類似した元を足す:

cos^{2} (x) + 9 cos^{2} (x) = 10 cos^{2} (x)

= 2 cos (x) + 10 cos^{2} (x) + 1 - 9

数を足す/引く:

1 - 9 = - 8

= 10 cos^{2} (x) + 2 cos (x) - 8

= 10 cos^{2} (x) + 2 cos (x) - 8

= 10 cos^{2} (x) + 2 cos (x) - 8

- 8 + 10 cos^{2} (x) + 2 cos (x) = 0

置換で解く

- 8 + 10 cos^{2} (x) + 2 cos (x) = 0

仮定:

cos (x) = u

- 8 + 10 u^{2} + 2 u = 0

- 8 + 10 u^{2} + 2 u = 0 : u = \frac{4}{5}, u = - 1

- 8 + 10 u^{2} + 2 u = 0

標準的な形式で書く

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

10 u^{2} + 2 u - 8 = 0

解くとthe二次式

10 u^{2} + 2 u - 8 = 0

二次Equationの公式:

次の場合:

a = 10, b = 2, c = - 8

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 10 ( - 8 )}{2 \cdot 10}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 10 ( - 8 )}{2 \cdot 10}

2^{2} - 4 \cdot 10 (- 8) = 18

2^{2} - 4 \cdot 10 (- 8)

規則を適用

- (- a) = a

= 2^{2} + 4 \cdot 10 \cdot 8

数を乗じる:

4 \cdot 10 \cdot 8 = 320

= 2^{2} + 320

2^{2} = 4

= 4 + 320

数を足す:

4 + 320 = 324

= 324

数を因数に分解する:

324 = 1 8^{2}

= 1 8^{2}

累乗根の規則を適用する:

n a^{n} = a

1 8^{2} = 18

= 18

u_{1, 2} = \frac{- 2 \pm 18}{2 \cdot 10}

解を分離する

u_{1} = \frac{- 2 + 18}{2 \cdot 10}, u_{2} = \frac{- 2 - 18}{2 \cdot 10}

u = \frac{- 2 + 18}{2 \cdot 10} : \frac{4}{5}

\frac{- 2 + 18}{2 \cdot 10}

数を足す/引く:

- 2 + 18 = 16

= \frac{16}{2 \cdot 10}

数を乗じる:

2 \cdot 10 = 20

= \frac{16}{20}

共通因数を約分する:

4

= \frac{4}{5}

u = \frac{- 2 - 18}{2 \cdot 10} : - 1

\frac{- 2 - 18}{2 \cdot 10}

数を引く:

- 2 - 18 = - 20

= \frac{- 20}{2 \cdot 10}

数を乗じる:

2 \cdot 10 = 20

= \frac{- 20}{20}

分数の規則を適用する:

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

= - \frac{20}{20}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

u = \frac{4}{5}, u = - 1

代用を戻す

u = cos (x)

cos (x) = \frac{4}{5}, cos (x) = - 1

cos (x) = \frac{4}{5}, cos (x) = - 1

cos (x) = \frac{4}{5} : x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

cos (x) = \frac{4}{5}

三角関数の逆数プロパティを適用する

cos (x) = \frac{4}{5}

以下の一般解

cos (x) = \frac{4}{5}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 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 = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 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) = 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

arccos (\frac{4}{5}) + 2 πn :

真

arccos (\frac{4}{5}) + 2 πn

挿入

n = 1

arccos (\frac{4}{5}) + 2 π 1

csc (x) + cot (x) = 3

の挿入向け

x = arccos (\frac{4}{5}) + 2 π 1

csc (arccos (\frac{4}{5}) + 2 π 1) + cot (arccos (\frac{4}{5}) + 2 π 1) = 3

改良

3 = 3

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{4}{5}) + 2 πn :

偽

2 π - arccos (\frac{4}{5}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{4}{5}) + 2 π 1

csc (x) + cot (x) = 3

の挿入向け

x = 2 π - arccos (\frac{4}{5}) + 2 π 1

csc (2 π - arccos (\frac{4}{5}) + 2 π 1) + cot (2 π - arccos (\frac{4}{5}) + 2 π 1) = 3

改良

- 3 = 3

\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) = 3

の挿入向け

x = π + 2 π 1

csc (π + 2 π 1) + cot (π + 2 π 1) = 3

未定義

\Rightarrow 偽

x = arccos (\frac{4}{5}) + 2 πn

10進法形式で解を証明する

x = 0.64350 \dots + 2 πn

グラフ

人気の例

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

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

sin(x/2)=(sqrt(3))/2

sin (\frac{x}{2}) = \frac{3}{2}

cos(x)tan(x)-cos(x)=0

cos (x) tan (x) - cos (x) = 0

4 cos (x + 7 0^{\circ}) = 3

4 cos (2 x) = 0