ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある三角関数 >

2tan^2(x)+cot^2(x)-3=0

解

2 tan^{2} (x) + cot^{2} (x) - 3 = 0

解

x = 0.61547 \dots + πn, x = 2.52611 \dots + πn, x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

+1

度

x = 35.26438 \dots^{\circ} + 18 0^{\circ} n, x = 144.73561 \dots^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

解答ステップ

2 tan^{2} (x) + cot^{2} (x) - 3 = 0

三角関数の公式を使用して書き換える

- 3 + cot^{2} (x) + 2 tan^{2} (x)

基本的な三角関数の公式を使用する:

tan (x) = \frac{1}{cot ( x )}

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

2 (\frac{1}{cot ( x )})^{2} = \frac{2}{cot ^{2} ( x )}

2 (\frac{1}{cot ( x )})^{2}

(\frac{1}{cot ( x )})^{2} = \frac{1}{cot ^{2} ( x )}

(\frac{1}{cot ( x )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{cot ^{2} ( x )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( x )}

= 2 \cdot \frac{1}{cot ^{2} ( x )}

分数を乗じる:

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

= \frac{1 \cdot 2}{cot ^{2} ( x )}

数を乗じる:

1 \cdot 2 = 2

= \frac{2}{cot ^{2} ( x )}

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

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

置換で解く

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

仮定:

cot (x) = u

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

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

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

以下で両辺を乗じる:

u^{2}

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

以下で両辺を乗じる:

u^{2}

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

簡素化

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

簡素化

u^{2} u^{2} : u^{4}

u^{2} u^{2}

指数の規則を適用する:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

数を足す:

2 + 2 = 4

= u^{4}

簡素化

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

\frac{2}{u ^{2}} u^{2}

分数を乗じる:

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

= \frac{2 u ^{2}}{u ^{2}}

共通因数を約分する:

u^{2}

= 2

簡素化

0 \cdot u^{2} : 0

0 \cdot u^{2}

規則を適用

0 \cdot a = 0

= 0

- 3 u^{2} + u^{4} + 2 = 0

- 3 u^{2} + u^{4} + 2 = 0

- 3 u^{2} + u^{4} + 2 = 0

解く

- 3 u^{2} + u^{4} + 2 = 0 : u = 2, u = - 2, u = 1, u = - 1

- 3 u^{2} + u^{4} + 2 = 0

標準的な形式で書く

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - 3 u^{2} + 2 = 0

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

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

解く

v^{2} - 3 v + 2 = 0 : v = 2, v = 1

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

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

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

数を乗じる:

4 \cdot 1 \cdot 2 = 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

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

解を分離する

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

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

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

規則を適用

- (- a) = a

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

数を足す:

3 + 1 = 4

= \frac{4}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{4}{2}

数を割る:

\frac{4}{2} = 2

= 2

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

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

規則を適用

- (- a) = a

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

数を引く:

3 - 1 = 2

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

数を乗じる:

2 \cdot 1 = 2

= \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

v = 2, v = 1

v = 2, v = 1

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

u^{2} = 2 : u = 2, u = - 2

u^{2} = 2

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = 2, u = - 2

解く

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

規則を適用

1 = 1

= 1

- 1 = - 1

- 1

規則を適用

1 = 1

= - 1

u = 1, u = - 1

解答は

u = 2, u = - 2, u = 1, u = - 1

u = 2, u = - 2, u = 1, u = - 1

解を検算する

未定義の (特異) 点を求める:

u = 0

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

の分母をゼロに比較する

解く

u^{2} = 0 : u = 0

u^{2} = 0

規則を適用

x^{n} = 0 \Rightarrow x = 0

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 = 2, u = - 2, u = 1, u = - 1

代用を戻す

u = cot (x)

cot (x) = 2, cot (x) = - 2, cot (x) = 1, cot (x) = - 1

cot (x) = 2, cot (x) = - 2, cot (x) = 1, cot (x) = - 1

cot (x) = 2 : x = arccot (2) + πn

cot (x) = 2

三角関数の逆数プロパティを適用する

cot (x) = 2

以下の一般解

cot (x) = 2

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (2) + πn

x = arccot (2) + πn

cot (x) = - 2 : x = arccot (- 2) + πn

cot (x) = - 2

三角関数の逆数プロパティを適用する

cot (x) = - 2

以下の一般解

cot (x) = - 2

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 2) + πn

x = arccot (- 2) + πn

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

以下の一般解

cot (x) = 1

cot (x)

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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

cot (x) = - 1

以下の一般解

cot (x) = - 1

cot (x)

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 0.61547 \dots + πn, x = 2.52611 \dots + πn, x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

グラフ

人気の例

((2sin(x)-1))/((sin(x)+5))=0

sec^2(b)=2+tan(b)

cos^{23}(x)+cos^2(x)=0

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