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人気のある三角関数 >

cos^2(2x)= 1/2-1/2 cos(2x)

解

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

解

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

+1

度

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

解答ステップ

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

置換で解く

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

仮定:

cos (2 x) = u

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

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

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

以下で両辺を乗じる:

2

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

以下で両辺を乗じる:

2

u^{2} \cdot 2 = \frac{1}{2} \cdot 2 - \frac{1}{2} u \cdot 2

簡素化

2 u^{2} = 1 - u

2 u^{2} = 1 - u

u

を左側に移動します

2 u^{2} = 1 - u

両辺に

u

を足す

2 u^{2} + u = 1 - u + u

簡素化

2 u^{2} + u = 1

2 u^{2} + u = 1

1

を左側に移動します

2 u^{2} + u = 1

両辺から

1

を引く

2 u^{2} + u - 1 = 1 - 1

簡素化

2 u^{2} + u - 1 = 0

2 u^{2} + u - 1 = 0

解くとthe二次式

2 u^{2} + u - 1 = 0

二次Equationの公式:

次の場合:

a = 2, b = 1, c = - 1

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

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

1^{2} - 4 \cdot 2 (- 1) = 3

1^{2} - 4 \cdot 2 (- 1)

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 1)

規則を適用

- (- a) = a

= 1 + 4 \cdot 2 \cdot 1

数を乗じる:

4 \cdot 2 \cdot 1 = 8

= 1 + 8

数を足す:

1 + 8 = 9

= 9

数を因数に分解する:

9 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 3

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

解を分離する

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

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

\frac{- 1 + 3}{2 \cdot 2}

数を足す/引く:

- 1 + 3 = 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{- 1 - 3}{2 \cdot 2} : - 1

\frac{- 1 - 3}{2 \cdot 2}

数を引く:

- 1 - 3 = - 4

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

数を乗じる:

2 \cdot 2 = 4

= \frac{- 4}{4}

分数の規則を適用する:

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

= - \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

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

代用を戻す

u = cos (2 x)

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

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

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) = - 1 : x = \frac{π}{2} + πn

cos (2 x) = - 1

以下の一般解

cos (2 x) = - 1

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 = π + 2 πn

2 x = π + 2 πn

解く

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

以下で両辺を割る

2

2 x = π + 2 πn

以下で両辺を割る

2

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

簡素化

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

すべての解を組み合わせる

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

グラフ

人気の例

2sin(A)-sqrt(3)=0

2 sin (A) - 3 = 0

sqrt(3)sec(3x)=2

3 sec (3 x) = 2

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

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2sin(3θ)=sqrt(3)

2 sin (3 θ) = 3

sin^2(x)+2sin(x)=3

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