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人気のある三角関数 >

2cos^2(2x)+3cos(2x)-2=0

解

2 cos^{2} (2 x) + 3 cos (2 x) - 2 = 0

解

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

+1

度

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

解答ステップ

2 cos^{2} (2 x) + 3 cos (2 x) - 2 = 0

置換で解く

2 cos^{2} (2 x) + 3 cos (2 x) - 2 = 0

仮定:

cos (2 x) = u

2 u^{2} + 3 u - 2 = 0

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

2 u^{2} + 3 u - 2 = 0

解くとthe二次式

2 u^{2} + 3 u - 2 = 0

二次Equationの公式:

次の場合:

a = 2, b = 3, c = - 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

3^{2} - 4 \cdot 2 (- 2) = 5

3^{2} - 4 \cdot 2 (- 2)

規則を適用

- (- a) = a

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

数を乗じる:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数を足す:

9 + 16 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

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

解を分離する

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

数を足す/引く:

- 3 + 5 = 2

= \frac{2}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{2}{4}

共通因数を約分する:

2

= \frac{1}{2}

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

\frac{- 3 - 5}{2 \cdot 2}

数を引く:

- 3 - 5 = - 8

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

数を乗じる:

2 \cdot 2 = 4

= \frac{- 8}{4}

分数の規則を適用する:

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

= - \frac{8}{4}

数を割る:

\frac{8}{4} = 2

= - 2

二次equationの解:

u = \frac{1}{2}, u = - 2

代用を戻す

u = cos (2 x)

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

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

cos (2 x) = \frac{1}{2} : x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

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

以下の一般解

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

2 x = \frac{π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

2 x = \frac{π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

解く

2 x = \frac{π}{3} + 2 πn : x = \frac{π}{6} + πn

2 x = \frac{π}{3} + 2 πn

以下で両辺を割る

2

2 x = \frac{π}{3} + 2 πn

以下で両辺を割る

2

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

簡素化

\frac{2 x}{2} = \frac{\frac{π}{3}}{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}{3}}{2} + \frac{2 πn}{2} : \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

分数の規則を適用する:

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

= \frac{π}{3 \cdot 2}

数を乗じる:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

解く

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{6} + πn

2 x = \frac{5 π}{3} + 2 πn

以下で両辺を割る

2

2 x = \frac{5 π}{3} + 2 πn

以下で両辺を割る

2

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

簡素化

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{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}{3}}{2} + \frac{2 πn}{2} : \frac{5 π}{6} + πn

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{3}}{2} = \frac{5 π}{6}

\frac{\frac{5 π}{3}}{2}

分数の規則を適用する:

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

= \frac{5 π}{3 \cdot 2}

数を乗じる:

3 \cdot 2 = 6

= \frac{5 π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

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

cos (2 x) = - 2 :

解なし

cos (2 x) = - 2

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

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

グラフ

人気の例

tan(t)=-12/5 ,(3pi)/2 <t<2pi

tan (t) = - \frac{12}{5}, \frac{3 π}{2} < t < 2 π

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