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人気のある三角関数 >

8cos(2x)+6=cos^2(x)+cos(x)

解

8 cos (2 x) + 6 = cos^{2} (x) + cos (x)

解

x = 1.15927 \dots + 2 πn, x = 2 π - 1.15927 \dots + 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

+1

度

x = 66.42182 \dots^{\circ} + 36 0^{\circ} n, x = 293.57817 \dots^{\circ} + 36 0^{\circ} n, x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

8 cos (2 x) + 6 = cos^{2} (x) + cos (x)

両辺から

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

を引く

8 cos (2 x) + 6 - cos^{2} (x) - cos (x) = 0

三角関数の公式を使用して書き換える

6 - cos (x) - cos^{2} (x) + 8 cos (2 x)

2倍角の公式を使用:

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

= 6 - cos (x) - cos^{2} (x) + 8 (2 cos^{2} (x) - 1)

簡素化

6 - cos (x) - cos^{2} (x) + 8 (2 cos^{2} (x) - 1) : 15 cos^{2} (x) - cos (x) - 2

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

拡張

8 (2 cos^{2} (x) - 1) : 16 cos^{2} (x) - 8

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

分配法則を適用する:

a (b - c) = ab - a c

a = 8, b = 2 cos^{2} (x), c = 1

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

簡素化

8 \cdot 2 cos^{2} (x) - 8 \cdot 1 : 16 cos^{2} (x) - 8

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

数を乗じる:

8 \cdot 2 = 16

= 16 cos^{2} (x) - 8 \cdot 1

数を乗じる:

8 \cdot 1 = 8

= 16 cos^{2} (x) - 8

= 16 cos^{2} (x) - 8

= 6 - cos (x) - cos^{2} (x) + 16 cos^{2} (x) - 8

簡素化

6 - cos (x) - cos^{2} (x) + 16 cos^{2} (x) - 8 : 15 cos^{2} (x) - cos (x) - 2

6 - cos (x) - cos^{2} (x) + 16 cos^{2} (x) - 8

類似した元を足す:

- cos^{2} (x) + 16 cos^{2} (x) = 15 cos^{2} (x)

= 6 - cos (x) + 15 cos^{2} (x) - 8

条件のようなグループ

= - cos (x) + 15 cos^{2} (x) + 6 - 8

数を足す/引く:

6 - 8 = - 2

= 15 cos^{2} (x) - cos (x) - 2

= 15 cos^{2} (x) - cos (x) - 2

= 15 cos^{2} (x) - cos (x) - 2

- 2 - cos (x) + 15 cos^{2} (x) = 0

置換で解く

- 2 - cos (x) + 15 cos^{2} (x) = 0

仮定:

cos (x) = u

- 2 - u + 15 u^{2} = 0

- 2 - u + 15 u^{2} = 0 : u = \frac{2}{5}, u = - \frac{1}{3}

- 2 - u + 15 u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 15, b = - 1, c = - 2

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 15 ( - 2 )}{2 \cdot 15}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 15 ( - 2 )}{2 \cdot 15}

(- 1)^{2} - 4 \cdot 15 (- 2) = 11

(- 1)^{2} - 4 \cdot 15 (- 2)

規則を適用

- (- a) = a

= (- 1)^{2} + 4 \cdot 15 \cdot 2

(- 1)^{2} = 1

(- 1)^{2}

指数の規則を適用する:

n

が偶数であれば

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

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

= 1^{2}

規則を適用

1^{a} = 1

= 1

4 \cdot 15 \cdot 2 = 120

4 \cdot 15 \cdot 2

数を乗じる:

4 \cdot 15 \cdot 2 = 120

= 120

= 1 + 120

数を足す:

1 + 120 = 121

= 121

数を因数に分解する:

121 = 1 1^{2}

= 1 1^{2}

累乗根の規則を適用する:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 1 ) \pm 11}{2 \cdot 15}

解を分離する

u_{1} = \frac{- ( - 1 ) + 11}{2 \cdot 15}, u_{2} = \frac{- ( - 1 ) - 11}{2 \cdot 15}

u = \frac{- ( - 1 ) + 11}{2 \cdot 15} : \frac{2}{5}

\frac{- ( - 1 ) + 11}{2 \cdot 15}

規則を適用

- (- a) = a

= \frac{1 + 11}{2 \cdot 15}

数を足す:

1 + 11 = 12

= \frac{12}{2 \cdot 15}

数を乗じる:

2 \cdot 15 = 30

= \frac{12}{30}

共通因数を約分する:

6

= \frac{2}{5}

u = \frac{- ( - 1 ) - 11}{2 \cdot 15} : - \frac{1}{3}

\frac{- ( - 1 ) - 11}{2 \cdot 15}

規則を適用

- (- a) = a

= \frac{1 - 11}{2 \cdot 15}

数を引く:

1 - 11 = - 10

= \frac{- 10}{2 \cdot 15}

数を乗じる:

2 \cdot 15 = 30

= \frac{- 10}{30}

分数の規則を適用する:

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

= - \frac{10}{30}

共通因数を約分する:

10

= - \frac{1}{3}

二次equationの解:

u = \frac{2}{5}, u = - \frac{1}{3}

代用を戻す

u = cos (x)

cos (x) = \frac{2}{5}, cos (x) = - \frac{1}{3}

cos (x) = \frac{2}{5}, cos (x) = - \frac{1}{3}

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

cos (x) = \frac{2}{5}

三角関数の逆数プロパティを適用する

cos (x) = \frac{2}{5}

以下の一般解

cos (x) = \frac{2}{5}

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

x = arccos (\frac{2}{5}) + 2 πn, x = 2 π - arccos (\frac{2}{5}) + 2 πn

x = arccos (\frac{2}{5}) + 2 πn, x = 2 π - arccos (\frac{2}{5}) + 2 πn

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

cos (x) = - \frac{1}{3}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{1}{3}

以下の一般解

cos (x) = - \frac{1}{3}

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

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

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

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 1.15927 \dots + 2 πn, x = 2 π - 1.15927 \dots + 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

グラフ

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