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人気のある三角関数 >

3cos(2x)=3sin(x)

解

3 cos (2 x) = 3 sin (x)

解

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = 27 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

解答ステップ

3 cos (2 x) = 3 sin (x)

両辺から

3 sin (x)

を引く

3 cos (2 x) - 3 sin (x) = 0

三角関数の公式を使用して書き換える

3 cos (2 x) - 3 sin (x)

2倍角の公式を使用:

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

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

(1 - 2 sin^{2} (x)) \cdot 3 - 3 sin (x) = 0

置換で解く

(1 - 2 sin^{2} (x)) \cdot 3 - 3 sin (x) = 0

仮定:

sin (x) = u

(1 - 2 u^{2}) \cdot 3 - 3 u = 0

(1 - 2 u^{2}) \cdot 3 - 3 u = 0 : u = - 1, u = \frac{1}{2}

(1 - 2 u^{2}) \cdot 3 - 3 u = 0

拡張

(1 - 2 u^{2}) \cdot 3 - 3 u : 3 - 6 u^{2} - 3 u

(1 - 2 u^{2}) \cdot 3 - 3 u

= 3 (1 - 2 u^{2}) - 3 u

拡張

3 (1 - 2 u^{2}) : 3 - 6 u^{2}

3 (1 - 2 u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = 1, c = 2 u^{2}

= 3 \cdot 1 - 3 \cdot 2 u^{2}

簡素化

3 \cdot 1 - 3 \cdot 2 u^{2} : 3 - 6 u^{2}

3 \cdot 1 - 3 \cdot 2 u^{2}

数を乗じる:

3 \cdot 1 = 3

= 3 - 3 \cdot 2 u^{2}

数を乗じる:

3 \cdot 2 = 6

= 3 - 6 u^{2}

= 3 - 6 u^{2}

= 3 - 6 u^{2} - 3 u

3 - 6 u^{2} - 3 u = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 6, b = - 3, c = 3

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

(- 3)^{2} - 4 (- 6) \cdot 3 = 9

(- 3)^{2} - 4 (- 6) \cdot 3

規則を適用

- (- a) = a

= (- 3)^{2} + 4 \cdot 6 \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 6 \cdot 3

数を乗じる:

4 \cdot 6 \cdot 3 = 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 ( - 6 )}

解を分離する

u_{1} = \frac{- ( - 3 ) + 9}{2 ( - 6 )}, u_{2} = \frac{- ( - 3 ) - 9}{2 ( - 6 )}

u = \frac{- ( - 3 ) + 9}{2 ( - 6 )} : - 1

\frac{- ( - 3 ) + 9}{2 ( - 6 )}

括弧を削除する:

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

= \frac{3 + 9}{- 2 \cdot 6}

数を足す:

3 + 9 = 12

= \frac{12}{- 2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{12}{- 12}

分数の規則を適用する:

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

= - \frac{12}{12}

規則を適用

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 3 ) - 9}{2 ( - 6 )} : \frac{1}{2}

\frac{- ( - 3 ) - 9}{2 ( - 6 )}

括弧を削除する:

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

= \frac{3 - 9}{- 2 \cdot 6}

数を引く:

3 - 9 = - 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 = - 1, u = \frac{1}{2}

代用を戻す

u = sin (x)

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

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

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

以下の一般解

sin (x) = - 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

以下の一般解

sin (x) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

すべての解を組み合わせる

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

人気の例

4cos(θ/2)+3=-2cos(θ)

4 cos (\frac{θ}{2}) + 3 = - 2 cos (θ)

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2 cos (2 x) + 3 = 23

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