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人気のある三角関数 >

2tan(x)=3csc(x)

解

2 tan (x) = 3 csc (x)

解

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

+1

度

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 tan (x) = 3 csc (x)

両辺から

3 csc (x)

を引く

2 tan (x) - 3 csc (x) = 0

サイン, コサインで表わす

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

簡素化

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

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

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

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

分数を乗じる:

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

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

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

3 \cdot \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 )}

数を乗じる:

1 \cdot 3 = 3

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

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

以下の最小公倍数:

cos (x), sin (x) : cos (x) sin (x)

cos (x), sin (x)

最小公倍数 (LCM)

cos (x)

または以下のいずれかに現れる因数で構成された式を計算する:

sin (x)

= cos (x) sin (x)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

cos (x) sin (x)

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

の場合

:

分母と分子に以下を乗じる:

sin (x)

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

\frac{3}{sin ( x )}

の場合

:

分母と分子に以下を乗じる:

cos (x)

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

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

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

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

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

両辺に

3 cos (x)

を足す

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

両辺を2乗する

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

両辺から

(3 cos (x))^{2}

を引く

4 sin^{4} (x) - 9 cos^{2} (x) = 0

因数

4 sin^{4} (x) - 9 cos^{2} (x) : (2 sin^{2} (x) + 3 cos (x)) (2 sin^{2} (x) - 3 cos (x))

4 sin^{4} (x) - 9 cos^{2} (x)

4 sin^{4} (x) - 9 cos^{2} (x)

を書き換え

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

4 sin^{4} (x) - 9 cos^{2} (x)

4

を書き換え

2^{2}

= 2^{2} sin^{4} (x) - 9 cos^{2} (x)

9

を書き換え

3^{2}

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

指数の規則を適用する:

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

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

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

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

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

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

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

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

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

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

2乗の差の公式を適用する:

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

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

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

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

各部分を別個に解く

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

2 sin^{2} (x) + 3 cos (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

三角関数の公式を使用して書き換える

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

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

(1 - cos^{2} (x)) \cdot 2 + 3 cos (x) = 0

置換で解く

(1 - cos^{2} (x)) \cdot 2 + 3 cos (x) = 0

仮定:

cos (x) = u

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

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

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

拡張

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

(1 - u^{2}) \cdot 2 + 3 u

= 2 (1 - u^{2}) + 3 u

拡張

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

数を乗じる:

2 \cdot 1 = 2

= 2 - 2 u^{2}

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

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

3^{2} - 4 (- 2) \cdot 2 = 5

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

規則を適用

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 2

数を乗じる:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数を足す:

9 + 16 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 ( - 2 )}

解を分離する

u_{1} = \frac{- 3 + 5}{2 ( - 2 )}, u_{2} = \frac{- 3 - 5}{2 ( - 2 )}

u = \frac{- 3 + 5}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 3 + 5}{2 ( - 2 )}

括弧を削除する:

(- a) = - a

= \frac{- 3 + 5}{- 2 \cdot 2}

数を足す/引く:

- 3 + 5 = 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}

u = \frac{- 3 - 5}{2 ( - 2 )} : 2

\frac{- 3 - 5}{2 ( - 2 )}

括弧を削除する:

(- a) = - a

= \frac{- 3 - 5}{- 2 \cdot 2}

数を引く:

- 3 - 5 = - 8

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

数を乗じる:

2 \cdot 2 = 4

= \frac{- 8}{- 4}

分数の規則を適用する:

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

= \frac{8}{4}

数を割る:

\frac{8}{4} = 2

= 2

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

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

2 sin^{2} (x) - 3 cos (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

三角関数の公式を使用して書き換える

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

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

(1 - cos^{2} (x)) \cdot 2 - 3 cos (x) = 0

置換で解く

(1 - cos^{2} (x)) \cdot 2 - 3 cos (x) = 0

仮定:

cos (x) = u

(1 - u^{2}) \cdot 2 - 3 u = 0

(1 - u^{2}) \cdot 2 - 3 u = 0 : u = - 2, u = \frac{1}{2}

(1 - u^{2}) \cdot 2 - 3 u = 0

拡張

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

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

= 2 (1 - u^{2}) - 3 u

拡張

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

数を乗じる:

2 \cdot 1 = 2

= 2 - 2 u^{2}

= 2 - 2 u^{2} - 3 u

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

(- 3)^{2} - 4 (- 2) \cdot 2 = 5

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

規則を適用

- (- a) = a

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

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 2

数を乗じる:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数を足す:

9 + 16 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 ( - 2 )}

解を分離する

u_{1} = \frac{- ( - 3 ) + 5}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 5}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 5}{2 ( - 2 )} : - 2

\frac{- ( - 3 ) + 5}{2 ( - 2 )}

括弧を削除する:

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

= \frac{3 + 5}{- 2 \cdot 2}

数を足す:

3 + 5 = 8

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

数を乗じる:

2 \cdot 2 = 4

= \frac{8}{- 4}

分数の規則を適用する:

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

= - \frac{8}{4}

数を割る:

\frac{8}{4} = 2

= - 2

u = \frac{- ( - 3 ) - 5}{2 ( - 2 )} : \frac{1}{2}

\frac{- ( - 3 ) - 5}{2 ( - 2 )}

括弧を削除する:

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

= \frac{3 - 5}{- 2 \cdot 2}

数を引く:

3 - 5 = - 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 = - 2, u = \frac{1}{2}

代用を戻す

u = cos (x)

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

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

cos (x) = - 2 :

解なし

cos (x) = - 2

- 1 \leq cos (x) \leq 1

解なし

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

すべての解を組み合わせる

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

元のequationに当てはめて解を検算する

2 tan (x) = 3 csc (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{2 π}{3} + 2 πn :

偽

\frac{2 π}{3} + 2 πn

挿入

n = 1

\frac{2 π}{3} + 2 π 1

2 tan (x) = 3 csc (x)

の挿入向け

x = \frac{2 π}{3} + 2 π 1

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

改良

- 3.46410 \dots = 3.46410 \dots

\Rightarrow 偽

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

2 tan (x) = 3 csc (x)

の挿入向け

x = \frac{4 π}{3} + 2 π 1

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

改良

3.46410 \dots = - 3.46410 \dots

\Rightarrow 偽

解答を確認する

\frac{π}{3} + 2 πn :

真

\frac{π}{3} + 2 πn

挿入

n = 1

\frac{π}{3} + 2 π 1

2 tan (x) = 3 csc (x)

の挿入向け

x = \frac{π}{3} + 2 π 1

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

改良

3.46410 \dots = 3.46410 \dots

\Rightarrow 真

解答を確認する

\frac{5 π}{3} + 2 πn :

真

\frac{5 π}{3} + 2 πn

挿入

n = 1

\frac{5 π}{3} + 2 π 1

2 tan (x) = 3 csc (x)

の挿入向け

x = \frac{5 π}{3} + 2 π 1

2 tan (\frac{5 π}{3} + 2 π 1) = 3 csc (\frac{5 π}{3} + 2 π 1)

改良

- 3.46410 \dots = - 3.46410 \dots

\Rightarrow 真

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

グラフ

人気の例

tan(θ)+1=sqrt(3)+sqrt(3)cot(θ)

tan (θ) + 1 = 3 + 3 cot (θ)

2-2cos^2(x)=2cos^2(x/2)

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

sin(x)+cos(x)=1.2,sin(2x)

sin (x) + cos (x) = 1.2, sin (2 x)

2cos(x)+3tan(x)=3sec(x)

2 cos (x) + 3 tan (x) = 3 sec (x)

14tan^2(x)=-14tan(x)

14 tan^{2} (x) = - 14 tan (x)