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人気のある三角関数 >

2sin^2(3x)-3cos(3x)+1=0

解

2 sin^{2} (3 x) - 3 cos (3 x) + 1 = 0

解

x = \frac{0.81462 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.81462 \dots}{3} + \frac{2 πn}{3}

+1

度

x = 15.55820 \dots^{\circ} + 12 0^{\circ} n, x = 104.44179 \dots^{\circ} + 12 0^{\circ} n

解答ステップ

2 sin^{2} (3 x) - 3 cos (3 x) + 1 = 0

三角関数の公式を使用して書き換える

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

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

簡素化

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

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

拡張

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

2 (1 - cos^{2} (3 x))

分配法則を適用する:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (3 x)

= 2 \cdot 1 - 2 cos^{2} (3 x)

数を乗じる:

2 \cdot 1 = 2

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

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

数を足す:

1 + 2 = 3

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

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

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

置換で解く

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

仮定:

cos (3 x) = u

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

3 - 2 u^{2} - 3 u = 0 : u = - \frac{3 + 33}{4}, u = \frac{33 - 3}{4}

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

(- 3)^{2} - 4 (- 2) \cdot 3 = 33

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

規則を適用

- (- a) = a

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

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 2 \cdot 3

数を乗じる:

4 \cdot 2 \cdot 3 = 24

= 3^{2} + 24

3^{2} = 9

= 9 + 24

数を足す:

9 + 24 = 33

= 33

u_{1, 2} = \frac{- ( - 3 ) \pm 33}{2 ( - 2 )}

解を分離する

u_{1} = \frac{- ( - 3 ) + 33}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 33}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 33}{2 ( - 2 )} : - \frac{3 + 33}{4}

\frac{- ( - 3 ) + 33}{2 ( - 2 )}

括弧を削除する:

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

= \frac{3 + 33}{- 2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{3 + 33}{- 4}

分数の規則を適用する:

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

= - \frac{3 + 33}{4}

u = \frac{- ( - 3 ) - 33}{2 ( - 2 )} : \frac{33 - 3}{4}

\frac{- ( - 3 ) - 33}{2 ( - 2 )}

括弧を削除する:

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

= \frac{3 - 33}{- 2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{3 - 33}{- 4}

分数の規則を適用する:

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

3 - 33 = - (33 - 3)

= \frac{33 - 3}{4}

二次equationの解:

u = - \frac{3 + 33}{4}, u = \frac{33 - 3}{4}

代用を戻す

u = cos (3 x)

cos (3 x) = - \frac{3 + 33}{4}, cos (3 x) = \frac{33 - 3}{4}

cos (3 x) = - \frac{3 + 33}{4}, cos (3 x) = \frac{33 - 3}{4}

cos (3 x) = - \frac{3 + 33}{4} :

解なし

cos (3 x) = - \frac{3 + 33}{4}

- 1 \leq cos (x) \leq 1

解なし

cos (3 x) = \frac{33 - 3}{4} : x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

cos (3 x) = \frac{33 - 3}{4}

三角関数の逆数プロパティを適用する

cos (3 x) = \frac{33 - 3}{4}

以下の一般解

cos (3 x) = \frac{33 - 3}{4}

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

3 x = arccos (\frac{33 - 3}{4}) + 2 πn, 3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

3 x = arccos (\frac{33 - 3}{4}) + 2 πn, 3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

解く

3 x = arccos (\frac{33 - 3}{4}) + 2 πn : x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

3 x = arccos (\frac{33 - 3}{4}) + 2 πn

以下で両辺を割る

3

3 x = arccos (\frac{33 - 3}{4}) + 2 πn

以下で両辺を割る

3

\frac{3 x}{3} = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

簡素化

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

解く

3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn : x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

以下で両辺を割る

3

3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

以下で両辺を割る

3

\frac{3 x}{3} = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

簡素化

x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

すべての解を組み合わせる

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

10進法形式で解を証明する

x = \frac{0.81462 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.81462 \dots}{3} + \frac{2 πn}{3}

グラフ

人気の例

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3 sin^{2} (x) + 6 sin (x) - 11 = 7 sin (x) - 9

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