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人気のある三角関数 >

2cos(2x)+3=4cos(x)

解

2 cos (2 x) + 3 = 4 cos (x)

解

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

+1

度

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 cos (2 x) + 3 = 4 cos (x)

両辺から

4 cos (x)

を引く

2 cos (2 x) + 3 - 4 cos (x) = 0

三角関数の公式を使用して書き換える

3 + 2 cos (2 x) - 4 cos (x)

2倍角の公式を使用:

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

= 3 + 2 (2 cos^{2} (x) - 1) - 4 cos (x)

簡素化

3 + 2 (2 cos^{2} (x) - 1) - 4 cos (x) : 4 cos^{2} (x) - 4 cos (x) + 1

3 + 2 (2 cos^{2} (x) - 1) - 4 cos (x)

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

簡素化

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

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

数を乗じる:

2 \cdot 2 = 4

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

数を乗じる:

2 \cdot 1 = 2

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

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

= 3 + 4 cos^{2} (x) - 2 - 4 cos (x)

簡素化

3 + 4 cos^{2} (x) - 2 - 4 cos (x) : 4 cos^{2} (x) - 4 cos (x) + 1

3 + 4 cos^{2} (x) - 2 - 4 cos (x)

条件のようなグループ

= 4 cos^{2} (x) - 4 cos (x) + 3 - 2

数を足す/引く:

3 - 2 = 1

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

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

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

1 - 4 cos (x) + 4 cos^{2} (x) = 0

置換で解く

1 - 4 cos (x) + 4 cos^{2} (x) = 0

仮定:

cos (x) = u

1 - 4 u + 4 u^{2} = 0

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

1 - 4 u + 4 u^{2} = 0

標準的な形式で書く

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

4 u^{2} - 4 u + 1 = 0

解くとthe二次式

4 u^{2} - 4 u + 1 = 0

二次Equationの公式:

次の場合:

a = 4, b = - 4, c = 1

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

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

(- 4)^{2} - 4 \cdot 4 \cdot 1 = 0

(- 4)^{2} - 4 \cdot 4 \cdot 1

指数の規則を適用する:

n

が偶数であれば

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

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

= 4^{2} - 4 \cdot 4 \cdot 1

数を乗じる:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

数を引く:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 \cdot 4}

u = \frac{- ( - 4 )}{2 \cdot 4}

\frac{- ( - 4 )}{2 \cdot 4} = \frac{1}{2}

\frac{- ( - 4 )}{2 \cdot 4}

規則を適用

- (- a) = a

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

数を乗じる:

2 \cdot 4 = 8

= \frac{4}{8}

共通因数を約分する:

4

= \frac{1}{2}

u = \frac{1}{2}

二次equationの解:

u = \frac{1}{2}

代用を戻す

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

以下の一般解

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

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

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

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

すべての解を組み合わせる

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

グラフ

人気の例

sin (2 θ) = 0.4

0.5=sin(x)cos(x)

0.5 = sin (x) cos (x)

sin(4x)-cos(4x)= 1/2

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

tan^2(θ)+sec(θ)=-2

tan^{2} (θ) + sec (θ) = - 2

9csc(x)=6sqrt(3)

9 csc (x) = 63