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人気のある三角関数 >

cos(2x)-7cos(x)+4=0

解

cos (2 x) - 7 cos (x) + 4 = 0

解

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

+1

度

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

解答ステップ

cos (2 x) - 7 cos (x) + 4 = 0

三角関数の公式を使用して書き換える

4 + cos (2 x) - 7 cos (x)

2倍角の公式を使用:

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

= 4 + 2 cos^{2} (x) - 1 - 7 cos (x)

簡素化

4 + 2 cos^{2} (x) - 1 - 7 cos (x) : 2 cos^{2} (x) - 7 cos (x) + 3

4 + 2 cos^{2} (x) - 1 - 7 cos (x)

条件のようなグループ

= 2 cos^{2} (x) - 7 cos (x) + 4 - 1

数を足す/引く:

4 - 1 = 3

= 2 cos^{2} (x) - 7 cos (x) + 3

= 2 cos^{2} (x) - 7 cos (x) + 3

3 + 2 cos^{2} (x) - 7 cos (x) = 0

置換で解く

3 + 2 cos^{2} (x) - 7 cos (x) = 0

仮定:

cos (x) = u

3 + 2 u^{2} - 7 u = 0

3 + 2 u^{2} - 7 u = 0 : u = 3, u = \frac{1}{2}

3 + 2 u^{2} - 7 u = 0

標準的な形式で書く

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

2 u^{2} - 7 u + 3 = 0

解くとthe二次式

2 u^{2} - 7 u + 3 = 0

二次Equationの公式:

次の場合:

a = 2, b = - 7, c = 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 \cdot 3 = 5

(- 7)^{2} - 4 \cdot 2 \cdot 3

指数の規則を適用する:

n

が偶数であれば

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

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 2 \cdot 3

数を乗じる:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

数を引く:

49 - 24 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 2}

解を分離する

u_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 2}

u = \frac{- ( - 7 ) + 5}{2 \cdot 2} : 3

\frac{- ( - 7 ) + 5}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{7 + 5}{2 \cdot 2}

数を足す:

7 + 5 = 12

= \frac{12}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{12}{4}

数を割る:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 7 ) - 5}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 7 ) - 5}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{7 - 5}{2 \cdot 2}

数を引く:

7 - 5 = 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}

二次equationの解:

u = 3, u = \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = 3, cos (x) = \frac{1}{2}

cos (x) = 3, cos (x) = \frac{1}{2}

cos (x) = 3 :

解なし

cos (x) = 3

- 1 \leq cos (x) \leq 1

解なし

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

グラフ

人気の例

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\frac{87.17}{sin ( 12 0 ^{\circ} )} = \frac{40}{sin ( x )}

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