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人気のある三角関数 >

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

解

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

解

x = π + 4 πn, x = 3 π + 4 πn, x = 2 \cdot 1.31811 \dots + 4 πn, x = 4 π - 2 \cdot 1.31811 \dots + 4 πn

+1

度

x = 18 0^{\circ} + 72 0^{\circ} n, x = 54 0^{\circ} + 72 0^{\circ} n, x = 151.04497 \dots^{\circ} + 72 0^{\circ} n, x = 568.95502 \dots^{\circ} + 72 0^{\circ} n

解答ステップ

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

仮定:

u = \frac{x}{2}

cos (u) - 2 cos (2 u) - 2 = 0

三角関数の公式を使用して書き換える

- 2 + cos (u) - 2 cos (2 u)

2倍角の公式を使用:

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

= - 2 + cos (u) - 2 (2 cos^{2} (u) - 1)

簡素化

- 2 + cos (u) - 2 (2 cos^{2} (u) - 1) : cos (u) - 4 cos^{2} (u)

- 2 + cos (u) - 2 (2 cos^{2} (u) - 1)

拡張

- 2 (2 cos^{2} (u) - 1) : - 4 cos^{2} (u) + 2

- 2 (2 cos^{2} (u) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = - 2, b = 2 cos^{2} (u), c = 1

= - 2 \cdot 2 cos^{2} (u) - (- 2) \cdot 1

マイナス・プラスの規則を適用する

- (- a) = a

= - 2 \cdot 2 cos^{2} (u) + 2 \cdot 1

簡素化

- 2 \cdot 2 cos^{2} (u) + 2 \cdot 1 : - 4 cos^{2} (u) + 2

- 2 \cdot 2 cos^{2} (u) + 2 \cdot 1

数を乗じる:

2 \cdot 2 = 4

= - 4 cos^{2} (u) + 2 \cdot 1

数を乗じる:

2 \cdot 1 = 2

= - 4 cos^{2} (u) + 2

= - 4 cos^{2} (u) + 2

= - 2 + cos (u) - 4 cos^{2} (u) + 2

簡素化

- 2 + cos (u) - 4 cos^{2} (u) + 2 : cos (u) - 4 cos^{2} (u)

- 2 + cos (u) - 4 cos^{2} (u) + 2

条件のようなグループ

= cos (u) - 4 cos^{2} (u) - 2 + 2

- 2 + 2 = 0

= cos (u) - 4 cos^{2} (u)

= cos (u) - 4 cos^{2} (u)

= cos (u) - 4 cos^{2} (u)

cos (u) - 4 cos^{2} (u) = 0

置換で解く

cos (u) - 4 cos^{2} (u) = 0

仮定:

cos (u) = u

u - 4 u^{2} = 0

u - 4 u^{2} = 0 : u = 0, u = \frac{1}{4}

u - 4 u^{2} = 0

標準的な形式で書く

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

- 4 u^{2} + u = 0

解くとthe二次式

- 4 u^{2} + u = 0

二次Equationの公式:

次の場合:

a = - 4, b = 1, c = 0

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

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

1^{2} - 4 (- 4) \cdot 0 = 1

1^{2} - 4 (- 4) \cdot 0

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 4) \cdot 0

規則を適用

- (- a) = a

= 1 + 4 \cdot 4 \cdot 0

規則を適用

0 \cdot a = 0

= 1 + 0

数を足す:

1 + 0 = 1

= 1

規則を適用

1 = 1

= 1

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

解を分離する

u_{1} = \frac{- 1 + 1}{2 ( - 4 )}, u_{2} = \frac{- 1 - 1}{2 ( - 4 )}

u = \frac{- 1 + 1}{2 ( - 4 )} : 0

\frac{- 1 + 1}{2 ( - 4 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 + 1}{- 2 \cdot 4}

数を足す/引く:

- 1 + 1 = 0

= \frac{0}{- 2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{0}{- 8}

分数の規則を適用する:

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

= - \frac{0}{8}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 1 - 1}{2 ( - 4 )} : \frac{1}{4}

\frac{- 1 - 1}{2 ( - 4 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 - 1}{- 2 \cdot 4}

数を引く:

- 1 - 1 = - 2

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

数を乗じる:

2 \cdot 4 = 8

= \frac{- 2}{- 8}

分数の規則を適用する:

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

= \frac{2}{8}

共通因数を約分する:

2

= \frac{1}{4}

二次equationの解:

u = 0, u = \frac{1}{4}

代用を戻す

u = cos (u)

cos (u) = 0, cos (u) = \frac{1}{4}

cos (u) = 0, cos (u) = \frac{1}{4}

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

以下の一般解

cos (u) = 0

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}

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = \frac{1}{4} : u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

cos (u) = \frac{1}{4}

三角関数の逆数プロパティを適用する

cos (u) = \frac{1}{4}

以下の一般解

cos (u) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

すべての解を組み合わせる

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

代用を戻す

u = \frac{x}{2}

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

\frac{x}{2} = \frac{π}{2} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{π}{2} + 2 πn

以下で両辺を乗じる:

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分数を乗じる:

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

= \frac{π 2}{2}

共通因数を約分する:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

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

\frac{x}{2} = \frac{3 π}{2} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

以下で両辺を乗じる:

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

分数を乗じる:

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

= \frac{3 π 2}{2}

共通因数を約分する:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

\frac{x}{2} = arccos (\frac{1}{4}) + 2 πn : x = 2 arccos (\frac{1}{4}) + 4 πn

\frac{x}{2} = arccos (\frac{1}{4}) + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = arccos (\frac{1}{4}) + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 arccos (\frac{1}{4}) + 2 \cdot 2 πn : 2 arccos (\frac{1}{4}) + 4 πn

2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 2 arccos (\frac{1}{4}) + 4 πn

x = 2 arccos (\frac{1}{4}) + 4 πn

x = 2 arccos (\frac{1}{4}) + 4 πn

x = 2 arccos (\frac{1}{4}) + 4 πn

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

\frac{x}{2} = 2 π - arccos (\frac{1}{4}) + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = 2 π - arccos (\frac{1}{4}) + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn : 4 π - 2 arccos (\frac{1}{4}) + 4 πn

2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

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

10進法形式で解を証明する

x = π + 4 πn, x = 3 π + 4 πn, x = 2 \cdot 1.31811 \dots + 4 πn, x = 4 π - 2 \cdot 1.31811 \dots + 4 πn

グラフ

人気の例

8*cos^2(3x)+6*cos(6x)=10216

8 \cdot cos^{2} (3 x) + 6 \cdot cos (6 x) = 10216

sin(x)=(48*sin(69))/(47.5)

sin (x) = \frac{48 \cdot sin ( 6 9 ^{\circ} )}{47.5}

cos(x)=(8sqrt(3))/(16)

cos (x) = \frac{8 3}{16}

solvefor x,cos^2(x)=0

so l v e f or x, cos^{2} (x) = 0

cos(θ)=(sqrt(7))/4

cos (θ) = \frac{7}{4}