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人気のある三角関数 >

sec(2x)+tan(2x)= 1/2

解

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

解

x = - \frac{0.64350 \dots}{2} + πn

+1

度

x = - 18.43494 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

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

両辺から

\frac{1}{2}

を引く

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

簡素化

sec (2 x) + tan (2 x) - \frac{1}{2} : \frac{2 sec ( 2 x ) + 2 tan ( 2 x ) - 1}{2}

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

元を分数に変換する:

sec (2 x) = \frac{sec ( 2 x ) 2}{2}, tan (2 x) = \frac{tan ( 2 x ) 2}{2}

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

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{sec ( 2 x ) \cdot 2 + tan ( 2 x ) \cdot 2 - 1}{2}

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

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

2 sec (2 x) + 2 tan (2 x) - 1 = 0

サイン, コサインで表わす

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

簡素化

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

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

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

2 \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 2}{cos ( 2 x )}

数を乗じる:

1 \cdot 2 = 2

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

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

2 \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 2}{cos ( 2 x )}

= \frac{2}{cos ( 2 x )} + \frac{2 sin ( 2 x )}{cos ( 2 x )} - 1

分数を組み合わせる

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

規則を適用

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

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

= \frac{2 sin ( 2 x ) + 2}{cos ( 2 x )} - 1

元を分数に変換する:

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

= \frac{2 + sin ( 2 x ) \cdot 2}{cos ( 2 x )} - \frac{1 \cdot 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{2 + sin ( 2 x ) \cdot 2 - 1 \cdot cos ( 2 x )}{cos ( 2 x )}

乗算:

1 \cdot cos (2 x) = cos (2 x)

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

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

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

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

両辺に

cos (2 x)

を足す

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

両辺を2乗する

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

両辺から

cos^{2} (2 x)

を引く

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

三角関数の公式を使用して書き換える

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

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

簡素化

(2 + 2 sin (2 x))^{2} - (1 - sin^{2} (2 x)) : 5 sin^{2} (2 x) + 8 sin (2 x) + 3

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

(2 + 2 sin (2 x))^{2} : 4 + 8 sin (2 x) + 4 sin^{2} (2 x)

完全平方式を適用する:

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

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

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

簡素化

2^{2} + 2 \cdot 2 \cdot 2 sin (2 x) + (2 sin (2 x))^{2} : 4 + 8 sin (2 x) + 4 sin^{2} (2 x)

2^{2} + 2 \cdot 2 \cdot 2 sin (2 x) + (2 sin (2 x))^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 \cdot 2 sin (2 x) = 8 sin (2 x)

2 \cdot 2 \cdot 2 sin (2 x)

数を乗じる:

2 \cdot 2 \cdot 2 = 8

= 8 sin (2 x)

(2 sin (2 x))^{2} = 4 sin^{2} (2 x)

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

指数の規則を適用する:

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

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

2^{2} = 4

= 4 sin^{2} (2 x)

= 4 + 8 sin (2 x) + 4 sin^{2} (2 x)

= 4 + 8 sin (2 x) + 4 sin^{2} (2 x)

= 4 + 8 sin (2 x) + 4 sin^{2} (2 x) - (1 - sin^{2} (2 x))

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

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

括弧を分配する

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

マイナス・プラスの規則を適用する

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

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

= 4 + 8 sin (2 x) + 4 sin^{2} (2 x) - 1 + sin^{2} (2 x)

簡素化

4 + 8 sin (2 x) + 4 sin^{2} (2 x) - 1 + sin^{2} (2 x) : 5 sin^{2} (2 x) + 8 sin (2 x) + 3

4 + 8 sin (2 x) + 4 sin^{2} (2 x) - 1 + sin^{2} (2 x)

条件のようなグループ

= 8 sin (2 x) + 4 sin^{2} (2 x) + sin^{2} (2 x) + 4 - 1

類似した元を足す:

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

= 8 sin (2 x) + 5 sin^{2} (2 x) + 4 - 1

数を足す/引く:

4 - 1 = 3

= 5 sin^{2} (2 x) + 8 sin (2 x) + 3

= 5 sin^{2} (2 x) + 8 sin (2 x) + 3

= 5 sin^{2} (2 x) + 8 sin (2 x) + 3

3 + 5 sin^{2} (2 x) + 8 sin (2 x) = 0

置換で解く

3 + 5 sin^{2} (2 x) + 8 sin (2 x) = 0

仮定:

sin (2 x) = u

3 + 5 u^{2} + 8 u = 0

3 + 5 u^{2} + 8 u = 0 : u = - \frac{3}{5}, u = - 1

3 + 5 u^{2} + 8 u = 0

標準的な形式で書く

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

5 u^{2} + 8 u + 3 = 0

解くとthe二次式

5 u^{2} + 8 u + 3 = 0

二次Equationの公式:

次の場合:

a = 5, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

8^{2} - 4 \cdot 5 \cdot 3 = 2

8^{2} - 4 \cdot 5 \cdot 3

数を乗じる:

4 \cdot 5 \cdot 3 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

数を引く:

64 - 60 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 8 \pm 2}{2 \cdot 5}

解を分離する

u_{1} = \frac{- 8 + 2}{2 \cdot 5}, u_{2} = \frac{- 8 - 2}{2 \cdot 5}

u = \frac{- 8 + 2}{2 \cdot 5} : - \frac{3}{5}

\frac{- 8 + 2}{2 \cdot 5}

数を足す/引く:

- 8 + 2 = - 6

= \frac{- 6}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 6}{10}

分数の規則を適用する:

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

= - \frac{6}{10}

共通因数を約分する:

2

= - \frac{3}{5}

u = \frac{- 8 - 2}{2 \cdot 5} : - 1

\frac{- 8 - 2}{2 \cdot 5}

数を引く:

- 8 - 2 = - 10

= \frac{- 10}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 10}{10}

分数の規則を適用する:

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

= - \frac{10}{10}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

u = - \frac{3}{5}, u = - 1

代用を戻す

u = sin (2 x)

sin (2 x) = - \frac{3}{5}, sin (2 x) = - 1

sin (2 x) = - \frac{3}{5}, sin (2 x) = - 1

sin (2 x) = - \frac{3}{5} : x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

sin (2 x) = - \frac{3}{5}

三角関数の逆数プロパティを適用する

sin (2 x) = - \frac{3}{5}

以下の一般解

sin (2 x) = - \frac{3}{5}

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

2 x = arcsin (- \frac{3}{5}) + 2 πn, 2 x = π + arcsin (\frac{3}{5}) + 2 πn

2 x = arcsin (- \frac{3}{5}) + 2 πn, 2 x = π + arcsin (\frac{3}{5}) + 2 πn

解く

2 x = arcsin (- \frac{3}{5}) + 2 πn : x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = arcsin (- \frac{3}{5}) + 2 πn

簡素化

arcsin (- \frac{3}{5}) + 2 πn : - arcsin (\frac{3}{5}) + 2 πn

arcsin (- \frac{3}{5}) + 2 πn

次のプロパティを使用する:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{5}) = - arcsin (\frac{3}{5})

= - arcsin (\frac{3}{5}) + 2 πn

2 x = - arcsin (\frac{3}{5}) + 2 πn

以下で両辺を割る

2

2 x = - arcsin (\frac{3}{5}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

簡素化

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

解く

2 x = π + arcsin (\frac{3}{5}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = π + arcsin (\frac{3}{5}) + 2 πn

以下で両辺を割る

2

2 x = π + arcsin (\frac{3}{5}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

簡素化

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

sin (2 x) = - 1 : x = \frac{3 π}{4} + πn

sin (2 x) = - 1

以下の一般解

sin (2 x) = - 1

sin (x)

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = \frac{3 π}{2} + 2 πn

2 x = \frac{3 π}{2} + 2 πn

解く

2 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{4} + πn

2 x = \frac{3 π}{2} + 2 πn

以下で両辺を割る

2

2 x = \frac{3 π}{2} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} : \frac{3 π}{4} + πn

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

分数の規則を適用する:

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

= \frac{3 π}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{3 π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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 = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{3 π}{4} + πn

元のequationに当てはめて解を検算する

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

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{3}{5} )}{2} + πn :

真

- \frac{arcsin ( \frac{3}{5} )}{2} + πn

挿入

n = 1

- \frac{arcsin ( \frac{3}{5} )}{2} + π 1

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

の挿入向け

x = - \frac{arcsin ( \frac{3}{5} )}{2} + π 1

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

改良

0.5 = 0.5

\Rightarrow 真

解答を確認する

\frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn :

偽

\frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

挿入

n = 1

\frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + π 1

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

の挿入向け

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + π 1

sec (2 (\frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + π 1)) + tan (2 (\frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + π 1)) = \frac{1}{2}

改良

- 0.5 = 0.5

\Rightarrow 偽

解答を確認する

\frac{3 π}{4} + πn :

偽

\frac{3 π}{4} + πn

挿入

n = 1

\frac{3 π}{4} + π 1

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

の挿入向け

x = \frac{3 π}{4} + π 1

sec (2 (\frac{3 π}{4} + π 1)) + tan (2 (\frac{3 π}{4} + π 1)) = \frac{1}{2}

未定義

\Rightarrow 偽

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

10進法形式で解を証明する

x = - \frac{0.64350 \dots}{2} + πn

グラフ

人気の例

2/(tan(a)+cot(a))=2sin(a)

\frac{2}{tan ( a ) + cot ( a )} = 2 sin (a)

2sin^2(w)+3sin(w)+1=0

2 sin^{2} (w) + 3 sin (w) + 1 = 0

tan(x)-2tan(x)cos(x)=0

tan (x) - 2 tan (x) cos (x) = 0

- 7 cos (7 x) = 0

1 - 4 sin^{2} (x) = 0