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人気のある三角関数 >

4sin^2(x)-7cos(x)=2

解

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

解

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

+1

度

x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

2

を引く

4 sin^{2} (x) - 7 cos (x) - 2 = 0

三角関数の公式を使用して書き換える

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

簡素化

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

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

数を乗じる:

4 \cdot 1 = 4

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

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

数を足す/引く:

- 2 + 4 = 2

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

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

2 - 4 cos^{2} (x) - 7 cos (x) = 0

置換で解く

2 - 4 cos^{2} (x) - 7 cos (x) = 0

仮定:

cos (x) = u

2 - 4 u^{2} - 7 u = 0

2 - 4 u^{2} - 7 u = 0 : u = - 2, u = \frac{1}{4}

2 - 4 u^{2} - 7 u = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 4, b = - 7, c = 2

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

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

(- 7)^{2} - 4 (- 4) \cdot 2 = 9

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

規則を適用

- (- a) = a

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

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 4 \cdot 2

数を乗じる:

4 \cdot 4 \cdot 2 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

数を足す:

49 + 32 = 81

= 81

数を因数に分解する:

81 = 9^{2}

= 9^{2}

累乗根の規則を適用する:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 7 ) \pm 9}{2 ( - 4 )}

解を分離する

u_{1} = \frac{- ( - 7 ) + 9}{2 ( - 4 )}, u_{2} = \frac{- ( - 7 ) - 9}{2 ( - 4 )}

u = \frac{- ( - 7 ) + 9}{2 ( - 4 )} : - 2

\frac{- ( - 7 ) + 9}{2 ( - 4 )}

括弧を削除する:

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

= \frac{7 + 9}{- 2 \cdot 4}

数を足す:

7 + 9 = 16

= \frac{16}{- 2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{16}{- 8}

分数の規則を適用する:

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

= - \frac{16}{8}

数を割る:

\frac{16}{8} = 2

= - 2

u = \frac{- ( - 7 ) - 9}{2 ( - 4 )} : \frac{1}{4}

\frac{- ( - 7 ) - 9}{2 ( - 4 )}

括弧を削除する:

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

= \frac{7 - 9}{- 2 \cdot 4}

数を引く:

7 - 9 = - 2

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

数を乗じる:

2 \cdot 4 = 8

= \frac{- 2}{- 8}

分数の規則を適用する:

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

= \frac{2}{8}

共通因数を約分する:

2

= \frac{1}{4}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = - 2 :

解なし

cos (x) = - 2

- 1 \leq cos (x) \leq 1

解なし

cos (x) = \frac{1}{4} : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

cos (x) = \frac{1}{4}

三角関数の逆数プロパティを適用する

cos (x) = \frac{1}{4}

以下の一般解

cos (x) = \frac{1}{4}

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

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

すべての解を組み合わせる

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

10進法形式で解を証明する

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

グラフ

人気の例

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