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人気のある三角関数 >

(1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131)

解

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

解

x = 2 \cdot 0.32845 \dots + 2 πn

+1

度

x = 37.63851 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

置換で解く

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

仮定:

tan (\frac{x}{2}) = u

\frac{1 + u}{1 - u} = 4.137131

\frac{1 + u}{1 - u} = 4.137131 : u = \frac{5.137131 - 2 4.137131}{3.137131}

\frac{1 + u}{1 - u} = 4.137131

以下で両辺を乗じる:

1 - u

\frac{1 + u}{1 - u} = 4.137131

以下で両辺を乗じる:

1 - u

\frac{1 + u}{1 - u} (1 - u) = 4.137131 (1 - u)

簡素化

1 + u = 4.137131 (1 - u)

1 + u = 4.137131 (1 - u)

拡張

4.137131 (1 - u) : 4.137131 - 4.137131 u

4.137131 (1 - u)

分配法則を適用する:

a (b - c) = ab - a c

a = 4.137131, b = 1, c = u

= 4.137131 \cdot 1 - 4.137131 u

= 1 \cdot 4.137131 - 4.137131 u

乗算:

1 \cdot 4.137131 = 4.137131

= 4.137131 - 4.137131 u

1 + u = 4.137131 - 4.137131 u

1

を右側に移動します

1 + u = 4.137131 - 4.137131 u

両辺から

1

を引く

1 + u - 1 = 4.137131 - 4.137131 u - 1

簡素化

u = 4.137131 - 4.137131 u - 1

u = 4.137131 - 4.137131 u - 1

4.137131 u

を左側に移動します

u = 4.137131 - 4.137131 u - 1

両辺に

4.137131 u

を足す

u + 4.137131 u = 4.137131 - 4.137131 u - 1 + 4.137131 u

簡素化

u + 4.137131 u = 4.137131 - 1

u + 4.137131 u = 4.137131 - 1

因数

u + 4.137131 u : (1 + 4.137131) u

u + 4.137131 u

共通項をくくり出す

u

= u (1 + 4.137131)

(1 + 4.137131) u = 4.137131 - 1

以下で両辺を割る

1 + 4.137131

(1 + 4.137131) u = 4.137131 - 1

以下で両辺を割る

1 + 4.137131

\frac{( 1 + 4.137131 ) u}{1 + 4.137131} = \frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131}

簡素化

\frac{( 1 + 4.137131 ) u}{1 + 4.137131} = \frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131}

簡素化

\frac{( 1 + 4.137131 ) u}{1 + 4.137131} : u

\frac{( 1 + 4.137131 ) u}{1 + 4.137131}

共通因数を約分する:

1 + 4.137131

= u

簡素化

\frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131} : \frac{5.137131 - 2 4.137131}{3.137131}

\frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131}

規則を適用

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

= \frac{4.137131 - 1}{1 + 4.137131}

共役で乗じる

\frac{1 - 4.137131}{1 - 4.137131}

= \frac{( 4.137131 - 1 ) ( 1 - 4.137131 )}{( 1 + 4.137131 ) ( 1 - 4.137131 )}

(4.137131 - 1) (1 - 4.137131) = 2 4.137131 - 5.137131

(4.137131 - 1) (1 - 4.137131)

FOIL メソッドを適用する:

(a + b) (c + d) = a c + a d + b c + b d

a = 4.137131, b = - 1, c = 1, d = - 4.137131

= 4.137131 \cdot 1 + 4.137131 (- 4.137131) + (- 1) \cdot 1 + (- 1) (- 4.137131)

マイナス・プラスの規則を適用する

+ (- a) = - a, (- a) (- b) = ab

= 1 \cdot 4.137131 - 4.137131 4.137131 - 1 \cdot 1 + 1 \cdot 4.137131

簡素化

1 \cdot 4.137131 - 4.137131 4.137131 - 1 \cdot 1 + 1 \cdot 4.137131 : 2 4.137131 - 5.137131

1 \cdot 4.137131 - 4.137131 4.137131 - 1 \cdot 1 + 1 \cdot 4.137131

類似した元を足す:

1 \cdot 4.137131 + 1 \cdot 4.137131 = 2 4.137131

= 2 4.137131 - 4.137131 4.137131 - 1 \cdot 1

累乗根の規則を適用する:

a a = a

4.137131 4.137131 = 4.137131

= 2 4.137131 - 4.137131 - 1 \cdot 1

数を乗じる:

1 \cdot 1 = 1

= 2 4.137131 - 4.137131 - 1

数を引く:

- 4.137131 - 1 = - 5.137131

= 2 4.137131 - 5.137131

= 2 4.137131 - 5.137131

(1 + 4.137131) (1 - 4.137131) = - 3.137131

(1 + 4.137131) (1 - 4.137131)

2乗の差の公式を適用する:

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

a = 1, b = 4.137131

= 1^{2} - (4.137131)^{2}

簡素化

1^{2} - (4.137131)^{2} : - 3.137131

1^{2} - (4.137131)^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - (4.137131)^{2}

(4.137131)^{2} = 4.137131

(4.137131)^{2}

累乗根の規則を適用する:

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

= (4.13713 1^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= 4.13713 1^{\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

= 4.137131

= 1 - 4.137131

数を引く:

1 - 4.137131 = - 3.137131

= - 3.137131

= - 3.137131

= \frac{2 4.137131 - 5.137131}{- 3.137131}

分数の規則を適用する:

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

2 4.137131 - 5.137131 = - (5.137131 - 2 4.137131)

= \frac{5.137131 - 2 4.137131}{3.137131}

u = \frac{5.137131 - 2 4.137131}{3.137131}

u = \frac{5.137131 - 2 4.137131}{3.137131}

u = \frac{5.137131 - 2 4.137131}{3.137131}

解を検算する

未定義の (特異) 点を求める:

u = 1

\frac{1 + u}{1 - u}

の分母をゼロに比較する

解く

1 - u = 0 : u = 1

1 - u = 0

1

を右側に移動します

1 - u = 0

両辺から

1

を引く

1 - u - 1 = 0 - 1

簡素化

- u = - 1

- u = - 1

以下で両辺を割る

- 1

- u = - 1

以下で両辺を割る

- 1

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

簡素化

u = 1

u = 1

以下の点は定義されていない

u = 1

未定義のポイントを解に組み合わせる:

u = \frac{5.137131 - 2 4.137131}{3.137131}

代用を戻す

u = tan (\frac{x}{2})

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131} : x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

三角関数の逆数プロパティを適用する

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

以下の一般解

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

tan (x) = a \Rightarrow x = arctan (a) + πn

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

解く

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn : x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

以下で両辺を乗じる:

2

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

簡素化

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

すべての解を組み合わせる

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

10進法形式で解を証明する

x = 2 \cdot 0.32845 \dots + 2 πn

グラフ

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