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人気のある三角関数 >

6sin^2(x)-5cos(x)-2=0

解

6 sin^{2} (x) - 5 cos (x) - 2 = 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

6 sin^{2} (x) - 5 cos (x) - 2 = 0

三角関数の公式を使用して書き換える

- 2 - 5 cos (x) + 6 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)

= - 2 - 5 cos (x) + 6 (1 - cos^{2} (x))

簡素化

- 2 - 5 cos (x) + 6 (1 - cos^{2} (x)) : - 6 cos^{2} (x) - 5 cos (x) + 4

- 2 - 5 cos (x) + 6 (1 - cos^{2} (x))

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

数を乗じる:

6 \cdot 1 = 6

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

= - 2 - 5 cos (x) + 6 - 6 cos^{2} (x)

簡素化

- 2 - 5 cos (x) + 6 - 6 cos^{2} (x) : - 6 cos^{2} (x) - 5 cos (x) + 4

- 2 - 5 cos (x) + 6 - 6 cos^{2} (x)

条件のようなグループ

= - 5 cos (x) - 6 cos^{2} (x) - 2 + 6

数を足す/引く:

- 2 + 6 = 4

= - 6 cos^{2} (x) - 5 cos (x) + 4

= - 6 cos^{2} (x) - 5 cos (x) + 4

= - 6 cos^{2} (x) - 5 cos (x) + 4

4 - 5 cos (x) - 6 cos^{2} (x) = 0

置換で解く

4 - 5 cos (x) - 6 cos^{2} (x) = 0

仮定:

cos (x) = u

4 - 5 u - 6 u^{2} = 0

4 - 5 u - 6 u^{2} = 0 : u = - \frac{4}{3}, u = \frac{1}{2}

4 - 5 u - 6 u^{2} = 0

標準的な形式で書く

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

- 6 u^{2} - 5 u + 4 = 0

解くとthe二次式

- 6 u^{2} - 5 u + 4 = 0

二次Equationの公式:

次の場合:

a = - 6, b = - 5, c = 4

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 4}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 4}{2 ( - 6 )}

(- 5)^{2} - 4 (- 6) \cdot 4 = 11

(- 5)^{2} - 4 (- 6) \cdot 4

規則を適用

- (- a) = a

= (- 5)^{2} + 4 \cdot 6 \cdot 4

指数の規則を適用する:

n

が偶数であれば

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

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 6 \cdot 4

数を乗じる:

4 \cdot 6 \cdot 4 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

数を足す:

25 + 96 = 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{- ( - 5 ) \pm 11}{2 ( - 6 )}

解を分離する

u_{1} = \frac{- ( - 5 ) + 11}{2 ( - 6 )}, u_{2} = \frac{- ( - 5 ) - 11}{2 ( - 6 )}

u = \frac{- ( - 5 ) + 11}{2 ( - 6 )} : - \frac{4}{3}

\frac{- ( - 5 ) + 11}{2 ( - 6 )}

括弧を削除する:

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

= \frac{5 + 11}{- 2 \cdot 6}

数を足す:

5 + 11 = 16

= \frac{16}{- 2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{16}{- 12}

分数の規則を適用する:

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

= - \frac{16}{12}

共通因数を約分する:

4

= - \frac{4}{3}

u = \frac{- ( - 5 ) - 11}{2 ( - 6 )} : \frac{1}{2}

\frac{- ( - 5 ) - 11}{2 ( - 6 )}

括弧を削除する:

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

= \frac{5 - 11}{- 2 \cdot 6}

数を引く:

5 - 11 = - 6

= \frac{- 6}{- 2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{- 6}{- 12}

分数の規則を適用する:

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

= \frac{6}{12}

共通因数を約分する:

6

= \frac{1}{2}

二次equationの解:

u = - \frac{4}{3}, u = \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = - \frac{4}{3}, cos (x) = \frac{1}{2}

cos (x) = - \frac{4}{3}, cos (x) = \frac{1}{2}

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

解なし

cos (x) = - \frac{4}{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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sin (x) = - cos (x), - 2 π \leq x \leq 2 π

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