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人気のある三角関数 >

solvefor x,7-3cos(x)=9sin^2(x)

解

解く

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

解

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

+1

度

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

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

両辺から

9 sin^{2} (x)

を引く

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

マイナス・プラスの規則を適用する

- (- a) = a

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

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

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

= 9 cos^{2} (x) - 3 cos (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{2}{3}, u = - \frac{1}{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 \cdot 9 ( - 2 )}{2 \cdot 9}

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

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

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

規則を適用

- (- a) = a

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

指数の規則を適用する:

n

が偶数であれば

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

(- 3)^{2} = 3^{2}

= 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 \cdot 9}

解を分離する

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

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

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

規則を適用

- (- a) = a

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

数を足す:

3 + 9 = 12

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

数を乗じる:

2 \cdot 9 = 18

= \frac{12}{18}

共通因数を約分する:

6

= \frac{2}{3}

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

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

規則を適用

- (- a) = a

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

数を引く:

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}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

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

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

グラフ

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