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人気のある三角関数 >

3sec^2(2x)=4

解

3 sec^{2} (2 x) = 4

解

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn, x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

+1

度

x = 1 5^{\circ} + 18 0^{\circ} n, x = 16 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n, x = 10 5^{\circ} + 18 0^{\circ} n

解答ステップ

3 sec^{2} (2 x) = 4

置換で解く

3 sec^{2} (2 x) = 4

仮定:

sec (2 x) = u

3 u^{2} = 4

3 u^{2} = 4 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

3 u^{2} = 4

以下で両辺を割る

3

3 u^{2} = 4

以下で両辺を割る

3

\frac{3 u ^{2}}{3} = \frac{4}{3}

簡素化

u^{2} = \frac{4}{3}

u^{2} = \frac{4}{3}

x^{2} = f (a)

の場合, 解は

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

u = \frac{4}{3}, u = - \frac{4}{3}

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

\frac{4}{3}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

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

= \frac{2}{3}

有理化する

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

共役で乗じる

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

累乗根の規則を適用する:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

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

- \frac{4}{3}

簡素化

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

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

= \frac{2}{3}

= - \frac{2}{3}

有理化する

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

共役で乗じる

\frac{3}{3}

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

33 = 3

33

累乗根の規則を適用する:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

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

代用を戻す

u = sec (2 x)

sec (2 x) = \frac{2 3}{3}, sec (2 x) = - \frac{2 3}{3}

sec (2 x) = \frac{2 3}{3}, sec (2 x) = - \frac{2 3}{3}

sec (2 x) = \frac{2 3}{3} : x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn

sec (2 x) = \frac{2 3}{3}

以下の一般解

sec (2 x) = \frac{2 3}{3}

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

解く

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{π}{6} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{π}{6}}{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{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

分数の規則を適用する:

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

= \frac{π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

解く

2 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{12} + πn

2 x = \frac{11 π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{11 π}{6} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{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{11 π}{6}}{2} + \frac{2 πn}{2} : \frac{11 π}{12} + πn

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{11 π}{6}}{2} = \frac{11 π}{12}

\frac{\frac{11 π}{6}}{2}

分数の規則を適用する:

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

= \frac{11 π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{11 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn

sec (2 x) = - \frac{2 3}{3} : x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

sec (2 x) = - \frac{2 3}{3}

以下の一般解

sec (2 x) = - \frac{2 3}{3}

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

2 x = \frac{5 π}{6} + 2 πn, 2 x = \frac{7 π}{6} + 2 πn

2 x = \frac{5 π}{6} + 2 πn, 2 x = \frac{7 π}{6} + 2 πn

解く

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{5 π}{6} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{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{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

分数の規則を適用する:

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

= \frac{5 π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

解く

2 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{12} + πn

2 x = \frac{7 π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{7 π}{6} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{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{7 π}{6}}{2} + \frac{2 πn}{2} : \frac{7 π}{12} + πn

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{6}}{2} = \frac{7 π}{12}

\frac{\frac{7 π}{6}}{2}

分数の規則を適用する:

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

= \frac{7 π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{7 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

すべての解を組み合わせる

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn, x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

グラフ

人気の例

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

arctan (x) = - 1

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9 cos (x) tan (x) = - 4 tan (x)

cos(θ)-sin(θ)=0

cos (θ) - sin (θ) = 0

tan^2(x)-9tan(x)-10=0

tan^{2} (x) - 9 tan (x) - 10 = 0