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人気のある三角関数 >

9sin^2(x)+3cos(x)-7=0

解

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

解

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

度

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

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

三角関数の公式を使用して書き換える

- 7 + 3 cos (x) + 9 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)

= - 7 + 3 cos (x) + 9 (1 - cos^{2} (x))

簡素化

- 7 + 3 cos (x) + 9 (1 - cos^{2} (x)) : 3 cos (x) - 9 cos^{2} (x) + 2

- 7 + 3 cos (x) + 9 (1 - cos^{2} (x))

拡張

9 (1 - cos^{2} (x)) : 9 - 9 cos^{2} (x)

9 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 9, b = 1, c = cos^{2} (x)

= 9 \cdot 1 - 9 cos^{2} (x)

数を乗じる:

9 \cdot 1 = 9

= 9 - 9 cos^{2} (x)

= - 7 + 3 cos (x) + 9 - 9 cos^{2} (x)

簡素化

- 7 + 3 cos (x) + 9 - 9 cos^{2} (x) : 3 cos (x) - 9 cos^{2} (x) + 2

- 7 + 3 cos (x) + 9 - 9 cos^{2} (x)

条件のようなグループ

= 3 cos (x) - 9 cos^{2} (x) - 7 + 9

数を足す/引く:

- 7 + 9 = 2

= 3 cos (x) - 9 cos^{2} (x) + 2

= 3 cos (x) - 9 cos^{2} (x) + 2

= 3 cos (x) - 9 cos^{2} (x) + 2

2 + 3 cos (x) - 9 cos^{2} (x) = 0

置換で解く

2 + 3 cos (x) - 9 cos^{2} (x) = 0

仮定:

cos (x) = u

2 + 3 u - 9 u^{2} = 0

2 + 3 u - 9 u^{2} = 0 : u = - \frac{1}{3}, u = \frac{2}{3}

2 + 3 u - 9 u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 9, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 9 ) \cdot 2}{2 ( - 9 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 9 ) \cdot 2}{2 ( - 9 )}

3^{2} - 4 (- 9) \cdot 2 = 9

3^{2} - 4 (- 9) \cdot 2

規則を適用

- (- a) = a

= 3^{2} + 4 \cdot 9 \cdot 2

数を乗じる:

4 \cdot 9 \cdot 2 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

数を足す:

9 + 72 = 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{- 3 \pm 9}{2 ( - 9 )}

解を分離する

u_{1} = \frac{- 3 + 9}{2 ( - 9 )}, u_{2} = \frac{- 3 - 9}{2 ( - 9 )}

u = \frac{- 3 + 9}{2 ( - 9 )} : - \frac{1}{3}

\frac{- 3 + 9}{2 ( - 9 )}

括弧を削除する:

(- a) = - a

= \frac{- 3 + 9}{- 2 \cdot 9}

数を足す/引く:

- 3 + 9 = 6

= \frac{6}{- 2 \cdot 9}

数を乗じる:

2 \cdot 9 = 18

= \frac{6}{- 18}

分数の規則を適用する:

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

= - \frac{6}{18}

共通因数を約分する:

6

= - \frac{1}{3}

u = \frac{- 3 - 9}{2 ( - 9 )} : \frac{2}{3}

\frac{- 3 - 9}{2 ( - 9 )}

括弧を削除する:

(- a) = - a

= \frac{- 3 - 9}{- 2 \cdot 9}

数を引く:

- 3 - 9 = - 12

= \frac{- 12}{- 2 \cdot 9}

数を乗じる:

2 \cdot 9 = 18

= \frac{- 12}{- 18}

分数の規則を適用する:

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

= \frac{12}{18}

共通因数を約分する:

6

= \frac{2}{3}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

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

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{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 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 = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

グラフ

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