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人気のある三角関数 >

8+4cos(2x)=12cos(x)

解

8 + 4 cos (2 x) = 12 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

8 + 4 cos (2 x) = 12 cos (x)

両辺から

12 cos (x)

を引く

8 + 4 cos (2 x) - 12 cos (x) = 0

三角関数の公式を使用して書き換える

8 - 12 cos (x) + 4 cos (2 x)

2倍角の公式を使用:

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

= 8 - 12 cos (x) + 4 (2 cos^{2} (x) - 1)

簡素化

8 - 12 cos (x) + 4 (2 cos^{2} (x) - 1) : 8 cos^{2} (x) - 12 cos (x) + 4

8 - 12 cos (x) + 4 (2 cos^{2} (x) - 1)

拡張

4 (2 cos^{2} (x) - 1) : 8 cos^{2} (x) - 4

4 (2 cos^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 4, b = 2 cos^{2} (x), c = 1

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

簡素化

4 \cdot 2 cos^{2} (x) - 4 \cdot 1 : 8 cos^{2} (x) - 4

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

数を乗じる:

4 \cdot 2 = 8

= 8 cos^{2} (x) - 4 \cdot 1

数を乗じる:

4 \cdot 1 = 4

= 8 cos^{2} (x) - 4

= 8 cos^{2} (x) - 4

= 8 - 12 cos (x) + 8 cos^{2} (x) - 4

簡素化

8 - 12 cos (x) + 8 cos^{2} (x) - 4 : 8 cos^{2} (x) - 12 cos (x) + 4

8 - 12 cos (x) + 8 cos^{2} (x) - 4

条件のようなグループ

= - 12 cos (x) + 8 cos^{2} (x) + 8 - 4

数を足す/引く:

8 - 4 = 4

= 8 cos^{2} (x) - 12 cos (x) + 4

= 8 cos^{2} (x) - 12 cos (x) + 4

= 8 cos^{2} (x) - 12 cos (x) + 4

4 - 12 cos (x) + 8 cos^{2} (x) = 0

置換で解く

4 - 12 cos (x) + 8 cos^{2} (x) = 0

仮定:

cos (x) = u

4 - 12 u + 8 u^{2} = 0

4 - 12 u + 8 u^{2} = 0 : u = 1, u = \frac{1}{2}

4 - 12 u + 8 u^{2} = 0

標準的な形式で書く

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

8 u^{2} - 12 u + 4 = 0

解くとthe二次式

8 u^{2} - 12 u + 4 = 0

二次Equationの公式:

次の場合:

a = 8, b = - 12, c = 4

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 8 \cdot 4}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 8 \cdot 4}{2 \cdot 8}

(- 12)^{2} - 4 \cdot 8 \cdot 4 = 4

(- 12)^{2} - 4 \cdot 8 \cdot 4

指数の規則を適用する:

n

が偶数であれば

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

(- 12)^{2} = 1 2^{2}

= 1 2^{2} - 4 \cdot 8 \cdot 4

数を乗じる:

4 \cdot 8 \cdot 4 = 128

= 1 2^{2} - 128

1 2^{2} = 144

= 144 - 128

数を引く:

144 - 128 = 16

= 16

数を因数に分解する:

16 = 4^{2}

= 4^{2}

累乗根の規則を適用する:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 12 ) \pm 4}{2 \cdot 8}

解を分離する

u_{1} = \frac{- ( - 12 ) + 4}{2 \cdot 8}, u_{2} = \frac{- ( - 12 ) - 4}{2 \cdot 8}

u = \frac{- ( - 12 ) + 4}{2 \cdot 8} : 1

\frac{- ( - 12 ) + 4}{2 \cdot 8}

規則を適用

- (- a) = a

= \frac{12 + 4}{2 \cdot 8}

数を足す:

12 + 4 = 16

= \frac{16}{2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{16}{16}

規則を適用

\frac{a}{a} = 1

= 1

u = \frac{- ( - 12 ) - 4}{2 \cdot 8} : \frac{1}{2}

\frac{- ( - 12 ) - 4}{2 \cdot 8}

規則を適用

- (- a) = a

= \frac{12 - 4}{2 \cdot 8}

数を引く:

12 - 4 = 8

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

数を乗じる:

2 \cdot 8 = 16

= \frac{8}{16}

共通因数を約分する:

8

= \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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