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人気のある三角関数 >

10sec(2x)+5tan(2x)-15=0

解

10 sec (2 x) + 5 tan (2 x) - 15 = 0

解

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

+1

度

x = 16.16676 \dots^{\circ} + 18 0^{\circ} n, x = - 34.60171 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

10 sec (2 x) + 5 tan (2 x) - 15 = 0

サイン, コサインで表わす

10 \cdot \frac{1}{cos ( 2 x )} + 5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 15 = 0

簡素化

10 \cdot \frac{1}{cos ( 2 x )} + 5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 15 : \frac{10 + 5 sin ( 2 x ) - 15 cos ( 2 x )}{cos ( 2 x )}

10 \cdot \frac{1}{cos ( 2 x )} + 5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 15

10 \cdot \frac{1}{cos ( 2 x )} = \frac{10}{cos ( 2 x )}

10 \cdot \frac{1}{cos ( 2 x )}

分数を乗じる:

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

= \frac{1 \cdot 10}{cos ( 2 x )}

数を乗じる:

1 \cdot 10 = 10

= \frac{10}{cos ( 2 x )}

5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} = \frac{5 sin ( 2 x )}{cos ( 2 x )}

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

分数を乗じる:

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

= \frac{sin ( 2 x ) \cdot 5}{cos ( 2 x )}

= \frac{10}{cos ( 2 x )} + \frac{5 sin ( 2 x )}{cos ( 2 x )} - 15

分数を組み合わせる

\frac{10}{cos ( 2 x )} + \frac{5 sin ( 2 x )}{cos ( 2 x )} : \frac{10 + 5 sin ( 2 x )}{cos ( 2 x )}

規則を適用

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

= \frac{10 + 5 sin ( 2 x )}{cos ( 2 x )}

= \frac{5 sin ( 2 x ) + 10}{cos ( 2 x )} - 15

元を分数に変換する:

15 = \frac{15 cos ( 2 x )}{cos ( 2 x )}

= \frac{10 + sin ( 2 x ) \cdot 5}{cos ( 2 x )} - \frac{15 cos ( 2 x )}{cos ( 2 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{10 + sin ( 2 x ) \cdot 5 - 15 cos ( 2 x )}{cos ( 2 x )}

\frac{10 + 5 sin ( 2 x ) - 15 cos ( 2 x )}{cos ( 2 x )} = 0

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

10 + 5 sin (2 x) - 15 cos (2 x) = 0

両辺に

15 cos (2 x)

を足す

10 + 5 sin (2 x) = 15 cos (2 x)

両辺を2乗する

(10 + 5 sin (2 x))^{2} = (15 cos (2 x))^{2}

両辺から

(15 cos (2 x))^{2}

を引く

(10 + 5 sin (2 x))^{2} - 225 cos^{2} (2 x) = 0

三角関数の公式を使用して書き換える

(10 + 5 sin (2 x))^{2} - 225 cos^{2} (2 x)

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

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

= (10 + 5 sin (2 x))^{2} - 225 (1 - sin^{2} (2 x))

簡素化

(10 + 5 sin (2 x))^{2} - 225 (1 - sin^{2} (2 x)) : 250 sin^{2} (2 x) + 100 sin (2 x) - 125

(10 + 5 sin (2 x))^{2} - 225 (1 - sin^{2} (2 x))

(10 + 5 sin (2 x))^{2} : 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

完全平方式を適用する:

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

a = 10, b = 5 sin (2 x)

= 1 0^{2} + 2 \cdot 10 \cdot 5 sin (2 x) + (5 sin (2 x))^{2}

簡素化

1 0^{2} + 2 \cdot 10 \cdot 5 sin (2 x) + (5 sin (2 x))^{2} : 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

1 0^{2} + 2 \cdot 10 \cdot 5 sin (2 x) + (5 sin (2 x))^{2}

1 0^{2} = 100

1 0^{2}

1 0^{2} = 100

= 100

2 \cdot 10 \cdot 5 sin (2 x) = 100 sin (2 x)

2 \cdot 10 \cdot 5 sin (2 x)

数を乗じる:

2 \cdot 10 \cdot 5 = 100

= 100 sin (2 x)

(5 sin (2 x))^{2} = 25 sin^{2} (2 x)

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

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

= 5^{2} sin^{2} (2 x)

5^{2} = 25

= 25 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 (1 - sin^{2} (2 x))

拡張

- 225 (1 - sin^{2} (2 x)) : - 225 + 225 sin^{2} (2 x)

- 225 (1 - sin^{2} (2 x))

分配法則を適用する:

a (b - c) = ab - a c

a = - 225, b = 1, c = sin^{2} (2 x)

= - 225 \cdot 1 - (- 225) sin^{2} (2 x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 225 \cdot 1 + 225 sin^{2} (2 x)

数を乗じる:

225 \cdot 1 = 225

= - 225 + 225 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 + 225 sin^{2} (2 x)

簡素化

100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 + 225 sin^{2} (2 x) : 250 sin^{2} (2 x) + 100 sin (2 x) - 125

100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 + 225 sin^{2} (2 x)

条件のようなグループ

= 100 sin (2 x) + 25 sin^{2} (2 x) + 225 sin^{2} (2 x) + 100 - 225

類似した元を足す:

25 sin^{2} (2 x) + 225 sin^{2} (2 x) = 250 sin^{2} (2 x)

= 100 sin (2 x) + 250 sin^{2} (2 x) + 100 - 225

数を足す/引く:

100 - 225 = - 125

= 250 sin^{2} (2 x) + 100 sin (2 x) - 125

= 250 sin^{2} (2 x) + 100 sin (2 x) - 125

= 250 sin^{2} (2 x) + 100 sin (2 x) - 125

- 125 + 100 sin (2 x) + 250 sin^{2} (2 x) = 0

置換で解く

- 125 + 100 sin (2 x) + 250 sin^{2} (2 x) = 0

仮定:

sin (2 x) = u

- 125 + 100 u + 250 u^{2} = 0

- 125 + 100 u + 250 u^{2} = 0 : u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

- 125 + 100 u + 250 u^{2} = 0

標準的な形式で書く

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

250 u^{2} + 100 u - 125 = 0

解くとthe二次式

250 u^{2} + 100 u - 125 = 0

二次Equationの公式:

次の場合:

a = 250, b = 100, c = - 125

u_{1, 2} = \frac{- 100 \pm 10 0 ^{2} - 4 \cdot 250 ( - 125 )}{2 \cdot 250}

u_{1, 2} = \frac{- 100 \pm 10 0 ^{2} - 4 \cdot 250 ( - 125 )}{2 \cdot 250}

10 0^{2} - 4 \cdot 250 (- 125) = 1506

10 0^{2} - 4 \cdot 250 (- 125)

規則を適用

- (- a) = a

= 10 0^{2} + 4 \cdot 250 \cdot 125

数を乗じる:

4 \cdot 250 \cdot 125 = 125000

= 10 0^{2} + 125000

10 0^{2} = 10000

= 10000 + 125000

数を足す:

10000 + 125000 = 135000

= 135000

以下の素因数分解:

135000 : 2^{3} \cdot 3^{3} \cdot 5^{4}

135000

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

指数の規則を適用する:

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

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

累乗根の規則を適用する:

n ab = n a n b

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

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

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

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

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

累乗根の規則を適用する:

n a^{m} = a^{\frac{m}{n}}

5^{4} = 5^{\frac{4}{2}} = 5^{2}

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

改良

= 1506

u_{1, 2} = \frac{- 100 \pm 150 6}{2 \cdot 250}

解を分離する

u_{1} = \frac{- 100 + 150 6}{2 \cdot 250}, u_{2} = \frac{- 100 - 150 6}{2 \cdot 250}

u = \frac{- 100 + 150 6}{2 \cdot 250} : \frac{- 2 + 3 6}{10}

\frac{- 100 + 150 6}{2 \cdot 250}

数を乗じる:

2 \cdot 250 = 500

= \frac{- 100 + 150 6}{500}

因数

- 100 + 1506 : 50 (- 2 + 36)

- 100 + 1506

書き換え

= - 50 \cdot 2 + 50 \cdot 36

共通項をくくり出す

50

= 50 (- 2 + 36)

= \frac{50 ( - 2 + 3 6 )}{500}

共通因数を約分する:

50

= \frac{- 2 + 3 6}{10}

u = \frac{- 100 - 150 6}{2 \cdot 250} : - \frac{2 + 3 6}{10}

\frac{- 100 - 150 6}{2 \cdot 250}

数を乗じる:

2 \cdot 250 = 500

= \frac{- 100 - 150 6}{500}

因数

- 100 - 1506 : - 50 (2 + 36)

- 100 - 1506

書き換え

= - 50 \cdot 2 - 50 \cdot 36

共通項をくくり出す

50

= - 50 (2 + 36)

= - \frac{50 ( 2 + 3 6 )}{500}

共通因数を約分する:

50

= - \frac{2 + 3 6}{10}

二次equationの解:

u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

代用を戻す

u = sin (2 x)

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10} : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = \frac{- 2 + 3 6}{10}

三角関数の逆数プロパティを適用する

sin (2 x) = \frac{- 2 + 3 6}{10}

以下の一般解

sin (2 x) = \frac{- 2 + 3 6}{10}

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

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, 2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, 2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

解く

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

簡素化

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

解く

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

簡素化

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10} : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10}

三角関数の逆数プロパティを適用する

sin (2 x) = - \frac{2 + 3 6}{10}

以下の一般解

sin (2 x) = - \frac{2 + 3 6}{10}

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

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, 2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, 2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

解く

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn

簡素化

arcsin (- \frac{2 + 3 6}{10}) + 2 πn : - arcsin (\frac{2 + 3 6}{10}) + 2 πn

arcsin (- \frac{2 + 3 6}{10}) + 2 πn

次のプロパティを使用する:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2 + 3 6}{10}) = - arcsin (\frac{2 + 3 6}{10})

= - arcsin (\frac{2 + 3 6}{10}) + 2 πn

2 x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

2 x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

簡素化

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

解く

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

簡素化

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

すべての解を組み合わせる

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

元のequationに当てはめて解を検算する

10 sec (2 x) + 5 tan (2 x) - 15 = 0

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{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn :

真

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

挿入

n = 1

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

10 sec (2 x) + 5 tan (2 x) - 15 = 0

の挿入向け

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

10 sec 2 \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 - 15 = 0

改良

0 = 0

\Rightarrow 真

解答を確認する

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn :

偽

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

挿入

n = 1

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

10 sec (2 x) + 5 tan (2 x) - 15 = 0

の挿入向け

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

10 sec 2 \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 - 15 = 0

改良

- 30 = 0

\Rightarrow 偽

解答を確認する

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn :

真

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

挿入

n = 1

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

10 sec (2 x) + 5 tan (2 x) - 15 = 0

の挿入向け

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

10 sec 2 - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 - 15 = 0

改良

0 = 0

\Rightarrow 真

解答を確認する

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn :

偽

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

挿入

n = 1

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

10 sec (2 x) + 5 tan (2 x) - 15 = 0

の挿入向け

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

10 sec 2 \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 - 15 = 0

改良

- 30 = 0

\Rightarrow 偽

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

10進法形式で解を証明する

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

グラフ

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