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人気のある三角関数 >

14+7cos(2x)=21cos(x)

解

14 + 7 cos (2 x) = 21 cos (x)

解

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

+1

度

x = 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

解答ステップ

14 + 7 cos (2 x) = 21 cos (x)

両辺から

21 cos (x)

を引く

14 + 7 cos (2 x) - 21 cos (x) = 0

三角関数の公式を使用して書き換える

14 - 21 cos (x) + 7 cos (2 x)

2倍角の公式を使用:

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

= 14 - 21 cos (x) + 7 (2 cos^{2} (x) - 1)

簡素化

14 - 21 cos (x) + 7 (2 cos^{2} (x) - 1) : 14 cos^{2} (x) - 21 cos (x) + 7

14 - 21 cos (x) + 7 (2 cos^{2} (x) - 1)

拡張

7 (2 cos^{2} (x) - 1) : 14 cos^{2} (x) - 7

7 (2 cos^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 7, b = 2 cos^{2} (x), c = 1

= 7 \cdot 2 cos^{2} (x) - 7 \cdot 1

簡素化

7 \cdot 2 cos^{2} (x) - 7 \cdot 1 : 14 cos^{2} (x) - 7

7 \cdot 2 cos^{2} (x) - 7 \cdot 1

数を乗じる:

7 \cdot 2 = 14

= 14 cos^{2} (x) - 7 \cdot 1

数を乗じる:

7 \cdot 1 = 7

= 14 cos^{2} (x) - 7

= 14 cos^{2} (x) - 7

= 14 - 21 cos (x) + 14 cos^{2} (x) - 7

簡素化

14 - 21 cos (x) + 14 cos^{2} (x) - 7 : 14 cos^{2} (x) - 21 cos (x) + 7

14 - 21 cos (x) + 14 cos^{2} (x) - 7

条件のようなグループ

= - 21 cos (x) + 14 cos^{2} (x) + 14 - 7

数を足す/引く:

14 - 7 = 7

= 14 cos^{2} (x) - 21 cos (x) + 7

= 14 cos^{2} (x) - 21 cos (x) + 7

= 14 cos^{2} (x) - 21 cos (x) + 7

7 + 14 cos^{2} (x) - 21 cos (x) = 0

置換で解く

7 + 14 cos^{2} (x) - 21 cos (x) = 0

仮定:

cos (x) = u

7 + 14 u^{2} - 21 u = 0

7 + 14 u^{2} - 21 u = 0 : u = 1, u = \frac{1}{2}

7 + 14 u^{2} - 21 u = 0

標準的な形式で書く

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

14 u^{2} - 21 u + 7 = 0

解くとthe二次式

14 u^{2} - 21 u + 7 = 0

二次Equationの公式:

次の場合:

a = 14, b = - 21, c = 7

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 \cdot 14 \cdot 7}{2 \cdot 14}

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 \cdot 14 \cdot 7}{2 \cdot 14}

(- 21)^{2} - 4 \cdot 14 \cdot 7 = 7

(- 21)^{2} - 4 \cdot 14 \cdot 7

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 21)^{2} = 2 1^{2}

= 2 1^{2} - 4 \cdot 14 \cdot 7

数を乗じる:

4 \cdot 14 \cdot 7 = 392

= 2 1^{2} - 392

2 1^{2} = 441

= 441 - 392

数を引く:

441 - 392 = 49

= 49

数を因数に分解する:

49 = 7^{2}

= 7^{2}

累乗根の規則を適用する:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 21 ) \pm 7}{2 \cdot 14}

解を分離する

u_{1} = \frac{- ( - 21 ) + 7}{2 \cdot 14}, u_{2} = \frac{- ( - 21 ) - 7}{2 \cdot 14}

u = \frac{- ( - 21 ) + 7}{2 \cdot 14} : 1

\frac{- ( - 21 ) + 7}{2 \cdot 14}

規則を適用

- (- a) = a

= \frac{21 + 7}{2 \cdot 14}

数を足す:

21 + 7 = 28

= \frac{28}{2 \cdot 14}

数を乗じる:

2 \cdot 14 = 28

= \frac{28}{28}

規則を適用

\frac{a}{a} = 1

= 1

u = \frac{- ( - 21 ) - 7}{2 \cdot 14} : \frac{1}{2}

\frac{- ( - 21 ) - 7}{2 \cdot 14}

規則を適用

- (- a) = a

= \frac{21 - 7}{2 \cdot 14}

数を引く:

21 - 7 = 14

= \frac{14}{2 \cdot 14}

数を乗じる:

2 \cdot 14 = 28

= \frac{14}{28}

共通因数を約分する:

14

= \frac{1}{2}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = 1 : x = 2 πn

cos (x) = 1

以下の一般解

cos (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}

x = 0 + 2 πn

x = 0 + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

以下の一般解

cos (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}

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

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

すべての解を組み合わせる

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

グラフ

人気の例

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sin (3 x) = 0, 0 \leq x < 2 π

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(sin(x))/(37)=(sin(30))/(30)

\frac{sin ( x )}{37} = \frac{sin ( 3 0 ^{\circ} )}{30}

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