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人気のある三角関数 >

2tan^2(x)-3cot^2(x)=5

解

2 tan^{2} (x) - 3 cot^{2} (x) = 5

解

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

+1

度

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

解答ステップ

2 tan^{2} (x) - 3 cot^{2} (x) = 5

両辺から

5

を引く

2 tan^{2} (x) - 3 cot^{2} (x) - 5 = 0

三角関数の公式を使用して書き換える

- 5 + 2 tan^{2} (x) - 3 cot^{2} (x)

基本的な三角関数の公式を使用する:

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

= - 5 + 2 (\frac{1}{cot ( x )})^{2} - 3 cot^{2} (x)

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 )}

= - 5 + \frac{2}{cot ^{2} ( x )} - 3 cot^{2} (x)

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

置換で解く

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

仮定:

cot (x) = u

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

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

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

以下で両辺を乗じる:

u^{2}

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

以下で両辺を乗じる:

u^{2}

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

簡素化

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

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

- 3 u^{2} u^{2} : - 3 u^{4}

- 3 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}

= - 3 u^{2 + 2}

数を足す:

2 + 2 = 4

= - 3 u^{4}

簡素化

0 \cdot u^{2} : 0

0 \cdot u^{2}

規則を適用

0 \cdot a = 0

= 0

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

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

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

解く

- 5 u^{2} + 2 - 3 u^{4} = 0 : u = 2 i, u = - 2 i, u = \frac{1}{3}, u = - \frac{1}{3}

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

標準的な形式で書く

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

- 3 u^{4} - 5 u^{2} + 2 = 0

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

- 3 v^{2} - 5 v + 2 = 0

解く

- 3 v^{2} - 5 v + 2 = 0 : v = - 2, v = \frac{1}{3}

- 3 v^{2} - 5 v + 2 = 0

解くとthe二次式

- 3 v^{2} - 5 v + 2 = 0

二次Equationの公式:

次の場合:

a = - 3, b = - 5, c = 2

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 3 ) \cdot 2}{2 ( - 3 )}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 3 ) \cdot 2}{2 ( - 3 )}

(- 5)^{2} - 4 (- 3) \cdot 2 = 7

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

規則を適用

- (- a) = a

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

指数の規則を適用する:

n

が偶数であれば

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

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 3 \cdot 2

数を乗じる:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数を足す:

25 + 24 = 49

= 49

数を因数に分解する:

49 = 7^{2}

= 7^{2}

累乗根の規則を適用する:

n a^{n} = a

7^{2} = 7

= 7

v_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 ( - 3 )}

解を分離する

v_{1} = \frac{- ( - 5 ) + 7}{2 ( - 3 )}, v_{2} = \frac{- ( - 5 ) - 7}{2 ( - 3 )}

v = \frac{- ( - 5 ) + 7}{2 ( - 3 )} : - 2

\frac{- ( - 5 ) + 7}{2 ( - 3 )}

括弧を削除する:

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

= \frac{5 + 7}{- 2 \cdot 3}

数を足す:

5 + 7 = 12

= \frac{12}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{12}{- 6}

分数の規則を適用する:

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

= - \frac{12}{6}

数を割る:

\frac{12}{6} = 2

= - 2

v = \frac{- ( - 5 ) - 7}{2 ( - 3 )} : \frac{1}{3}

\frac{- ( - 5 ) - 7}{2 ( - 3 )}

括弧を削除する:

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

= \frac{5 - 7}{- 2 \cdot 3}

数を引く:

5 - 7 = - 2

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

数を乗じる:

2 \cdot 3 = 6

= \frac{- 2}{- 6}

分数の規則を適用する:

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

= \frac{2}{6}

共通因数を約分する:

2

= \frac{1}{3}

二次equationの解:

v = - 2, v = \frac{1}{3}

v = - 2, v = \frac{1}{3}

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

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

u^{2} = - 2

x^{2} = f (a)

の場合, 解は

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

u = - 2, u = - - 2

簡素化

- 2 : 2 i

- 2

累乗根の規則を適用する:

- a = - 1 a

- 2 = - 1 2

= - 1 2

虚数の規則を適用する:

- 1 = i

= 2 i

簡素化

- - 2 : - 2 i

- - 2

簡素化

- 2 : 2 i

- 2

累乗根の規則を適用する:

- a = - 1 a

- 2 = - 1 2

= - 1 2

虚数の規則を適用する:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

解く

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

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

x^{2} = f (a)

の場合, 解は

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

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

解答は

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

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

解を検算する

未定義の (特異) 点を求める:

u = 0

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

代用を戻す

u = cot (x)

cot (x) = 2 i, cot (x) = - 2 i, cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = 2 i, cot (x) = - 2 i, cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = 2 i :

解なし

cot (x) = 2 i

解なし

cot (x) = - 2 i :

解なし

cot (x) = - 2 i

解なし

cot (x) = \frac{1}{3} : x = arccot (\frac{1}{3}) + πn

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

x = arccot (\frac{1}{3}) + πn

x = arccot (\frac{1}{3}) + πn

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

x = arccot (- \frac{1}{3}) + πn

x = arccot (- \frac{1}{3}) + πn

すべての解を組み合わせる

x = arccot (\frac{1}{3}) + πn, x = arccot (- \frac{1}{3}) + πn

10進法形式で解を証明する

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

グラフ

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