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人気のある三角関数 >

8tan(x/2)+8cos(x)tan(x/2)=1

解

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) = 1

解

x = 0.12532 \dots + 2 πn, x = π - 0.12532 \dots + 2 πn

+1

度

x = 7.18075 \dots^{\circ} + 36 0^{\circ} n, x = 172.81924 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) = 1

両辺から

1

を引く

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) - 1 = 0

仮定:

u = \frac{x}{2}

8 tan (u) + 8 cos (2 u) tan (u) - 1 = 0

三角関数の公式を使用して書き換える

- 1 + 8 tan (u) + 8 cos (2 u) tan (u)

2倍角の公式を使用:

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

= - 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

簡素化

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

= - 1 + 8 tan (u) + 8 tan (u) (2 cos^{2} (u) - 1)

拡張

8 tan (u) (2 cos^{2} (u) - 1) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 tan (u) (2 cos^{2} (u) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 8 tan (u), b = 2 cos^{2} (u), c = 1

= 8 tan (u) \cdot 2 cos^{2} (u) - 8 tan (u) \cdot 1

= 8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

簡素化

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

数を乗じる:

8 \cdot 2 = 16

= 16 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

数を乗じる:

8 \cdot 1 = 8

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= - 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

簡素化

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

条件のようなグループ

= 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) - 1

類似した元を足す:

8 tan (u) - 8 tan (u) = 0

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

基本的な三角関数の公式を使用する:

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

= - 1 + 16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )} = 16 sin (u) cos (u)

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

分数を乗じる:

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

= \frac{sin ( u ) \cdot 16 cos ^{2} ( u )}{cos ( u )}

共通因数を約分する:

cos (u)

= 16 sin (u) cos (u)

= - 1 + 16 sin (u) cos (u)

2倍角の公式を使用:

2 sin (x) cos (x) = sin (2 x)

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

= - 1 + 16 \cdot \frac{sin ( 2 u )}{2}

- 1 + 16 \cdot \frac{sin ( 2 u )}{2} = 0

16 \cdot \frac{sin ( 2 u )}{2} = 8 sin (2 u)

16 \cdot \frac{sin ( 2 u )}{2}

分数を乗じる:

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

= \frac{sin ( 2 u ) \cdot 16}{2}

数を割る:

\frac{16}{2} = 8

= 8 sin (2 u)

- 1 + 8 sin (2 u) = 0

1

を右側に移動します

- 1 + 8 sin (2 u) = 0

両辺に

1

を足す

- 1 + 8 sin (2 u) + 1 = 0 + 1

簡素化

8 sin (2 u) = 1

8 sin (2 u) = 1

以下で両辺を割る

8

8 sin (2 u) = 1

以下で両辺を割る

8

\frac{8 sin ( 2 u )}{8} = \frac{1}{8}

簡素化

sin (2 u) = \frac{1}{8}

sin (2 u) = \frac{1}{8}

三角関数の逆数プロパティを適用する

sin (2 u) = \frac{1}{8}

以下の一般解

sin (2 u) = \frac{1}{8}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

解く

2 u = arcsin (\frac{1}{8}) + 2 πn : u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

2 u = arcsin (\frac{1}{8}) + 2 πn

以下で両辺を割る

2

2 u = arcsin (\frac{1}{8}) + 2 πn

以下で両辺を割る

2

\frac{2 u}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + \frac{2 πn}{2}

簡素化

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

解く

2 u = π - arcsin (\frac{1}{8}) + 2 πn : u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

2 u = π - arcsin (\frac{1}{8}) + 2 πn

以下で両辺を割る

2

2 u = π - arcsin (\frac{1}{8}) + 2 πn

以下で両辺を割る

2

\frac{2 u}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + \frac{2 πn}{2}

簡素化

u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn, u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

代用を戻す

u = \frac{x}{2}

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn : x = arcsin (\frac{1}{8}) + 2 πn

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn : arcsin (\frac{1}{8}) + 2 πn

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} = arcsin (\frac{1}{8})

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2}

分数を乗じる:

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

= \frac{arcsin ( \frac{1}{8} ) \cdot 2}{2}

共通因数を約分する:

2

= arcsin (\frac{1}{8})

= arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn

\frac{x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn : x = π - arcsin (\frac{1}{8}) + 2 πn

\frac{x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn : π - arcsin (\frac{1}{8}) + 2 πn

2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分数を乗じる:

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

= \frac{π 2}{2}

共通因数を約分する:

2

= π

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} = arcsin (\frac{1}{8})

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2}

分数を乗じる:

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

= \frac{arcsin ( \frac{1}{8} ) \cdot 2}{2}

共通因数を約分する:

2

= arcsin (\frac{1}{8})

= π - arcsin (\frac{1}{8}) + 2 πn

x = π - arcsin (\frac{1}{8}) + 2 πn

x = π - arcsin (\frac{1}{8}) + 2 πn

x = π - arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn, x = π - arcsin (\frac{1}{8}) + 2 πn

10進法形式で解を証明する

x = 0.12532 \dots + 2 πn, x = π - 0.12532 \dots + 2 πn

グラフ

人気の例

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