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人気のある三角関数 >

sqrt(2)cos(x)-1=cos(2x)

解

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

解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

cos (2 x)

を引く

2 cos (x) - 1 - cos (2 x) = 0

三角関数の公式を使用して書き換える

- 1 - cos (2 x) + cos (x) 2

2倍角の公式を使用:

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

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

簡素化

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

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

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

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

括弧を分配する

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

マイナス・プラスの規則を適用する

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

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

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

簡素化

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

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

条件のようなグループ

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

- 1 + 1 = 0

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

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

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

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

置換で解く

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

仮定:

cos (x) = u

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

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

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 2, b = 2, c = 0

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

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

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

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

規則を適用

- (- a) = a

= (2)^{2} + 4 \cdot 2 \cdot 0

(2)^{2} = 2

(2)^{2}

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 2

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

規則を適用

0 \cdot a = 0

= 0

= 2 + 0

数を足す:

2 + 0 = 2

= 2

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

解を分離する

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

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

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

括弧を削除する:

(- a) = - a

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

類似した元を足す:

- 2 + 2 = 0

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

数を乗じる:

2 \cdot 2 = 4

= \frac{0}{- 4}

分数の規則を適用する:

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

= - \frac{0}{4}

規則を適用

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

= - 0

= 0

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

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

括弧を削除する:

(- a) = - a

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

類似した元を足す:

- 2 - 2 = - 22

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

数を乗じる:

2 \cdot 2 = 4

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

分数の規則を適用する:

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

= \frac{2 2}{4}

共通因数を約分する:

2

= \frac{2}{2}

二次equationの解:

u = 0, u = \frac{2}{2}

代用を戻す

u = cos (x)

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

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

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

cos (x) = 0

以下の一般解

cos (x) = 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}

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

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

cos (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

以下の一般解

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

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

すべての解を組み合わせる

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

グラフ

人気の例

0 = 4 cos (2 θ)

tan^2(x)cos(x)=tan^2(x)

tan^{2} (x) cos (x) = tan^{2} (x)

cos(x)sin(x)=3sin(x)

cos (x) sin (x) = 3 sin (x)

tan^3(x)-tan(x)=0

tan^{3} (x) - tan (x) = 0

8 sin (θ) - 1 = 0