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人気のある三角関数 >

12sin^2(x)-cos(x)-6=0

解

12 sin^{2} (x) - cos (x) - 6 = 0

解

x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

度

x = 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = - 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

12 sin^{2} (x) - cos (x) - 6 = 0

三角関数の公式を使用して書き換える

- 6 - cos (x) + 12 sin^{2} (x)

ピタゴラスの公式を使用する:

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

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

= - 6 - cos (x) + 12 (1 - cos^{2} (x))

簡素化

- 6 - cos (x) + 12 (1 - cos^{2} (x)) : - 12 cos^{2} (x) - cos (x) + 6

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

拡張

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

12 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 12, b = 1, c = cos^{2} (x)

= 12 \cdot 1 - 12 cos^{2} (x)

数を乗じる:

12 \cdot 1 = 12

= 12 - 12 cos^{2} (x)

= - 6 - cos (x) + 12 - 12 cos^{2} (x)

簡素化

- 6 - cos (x) + 12 - 12 cos^{2} (x) : - 12 cos^{2} (x) - cos (x) + 6

- 6 - cos (x) + 12 - 12 cos^{2} (x)

条件のようなグループ

= - cos (x) - 12 cos^{2} (x) - 6 + 12

数を足す/引く:

- 6 + 12 = 6

= - 12 cos^{2} (x) - cos (x) + 6

= - 12 cos^{2} (x) - cos (x) + 6

= - 12 cos^{2} (x) - cos (x) + 6

6 - cos (x) - 12 cos^{2} (x) = 0

置換で解く

6 - cos (x) - 12 cos^{2} (x) = 0

仮定:

cos (x) = u

6 - u - 12 u^{2} = 0

6 - u - 12 u^{2} = 0 : u = - \frac{3}{4}, u = \frac{2}{3}

6 - u - 12 u^{2} = 0

標準的な形式で書く

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

- 12 u^{2} - u + 6 = 0

解くとthe二次式

- 12 u^{2} - u + 6 = 0

二次Equationの公式:

次の場合:

a = - 12, b = - 1, c = 6

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 12 ) \cdot 6}{2 ( - 12 )}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 12 ) \cdot 6}{2 ( - 12 )}

(- 1)^{2} - 4 (- 12) \cdot 6 = 17

(- 1)^{2} - 4 (- 12) \cdot 6

規則を適用

- (- a) = a

= (- 1)^{2} + 4 \cdot 12 \cdot 6

(- 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 12 \cdot 6 = 288

4 \cdot 12 \cdot 6

数を乗じる:

4 \cdot 12 \cdot 6 = 288

= 288

= 1 + 288

数を足す:

1 + 288 = 289

= 289

数を因数に分解する:

289 = 1 7^{2}

= 1 7^{2}

累乗根の規則を適用する:

n a^{n} = a

1 7^{2} = 17

= 17

u_{1, 2} = \frac{- ( - 1 ) \pm 17}{2 ( - 12 )}

解を分離する

u_{1} = \frac{- ( - 1 ) + 17}{2 ( - 12 )}, u_{2} = \frac{- ( - 1 ) - 17}{2 ( - 12 )}

u = \frac{- ( - 1 ) + 17}{2 ( - 12 )} : - \frac{3}{4}

\frac{- ( - 1 ) + 17}{2 ( - 12 )}

括弧を削除する:

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

= \frac{1 + 17}{- 2 \cdot 12}

数を足す:

1 + 17 = 18

= \frac{18}{- 2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{18}{- 24}

分数の規則を適用する:

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

= - \frac{18}{24}

共通因数を約分する:

6

= - \frac{3}{4}

u = \frac{- ( - 1 ) - 17}{2 ( - 12 )} : \frac{2}{3}

\frac{- ( - 1 ) - 17}{2 ( - 12 )}

括弧を削除する:

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

= \frac{1 - 17}{- 2 \cdot 12}

数を引く:

1 - 17 = - 16

= \frac{- 16}{- 2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{- 16}{- 24}

分数の規則を適用する:

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

= \frac{16}{24}

共通因数を約分する:

8

= \frac{2}{3}

二次equationの解:

u = - \frac{3}{4}, u = \frac{2}{3}

代用を戻す

u = cos (x)

cos (x) = - \frac{3}{4}, cos (x) = \frac{2}{3}

cos (x) = - \frac{3}{4}, cos (x) = \frac{2}{3}

cos (x) = - \frac{3}{4} : x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

cos (x) = - \frac{3}{4}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{3}{4}

以下の一般解

cos (x) = - \frac{3}{4}

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

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

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

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

すべての解を組み合わせる

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

10進法形式で解を証明する

x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

グラフ

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