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人気のある三角関数 >

4cos(θ/2)+3=-2cos(θ)

解

4 cos (\frac{θ}{2}) + 3 = - 2 cos (θ)

解

θ = \frac{4 π}{3} + 4 πn, θ = \frac{8 π}{3} + 4 πn

+1

度

θ = 24 0^{\circ} + 72 0^{\circ} n, θ = 48 0^{\circ} + 72 0^{\circ} n

解答ステップ

4 cos (\frac{θ}{2}) + 3 = - 2 cos (θ)

両辺から

- 2 cos (θ)

を引く

4 cos (\frac{θ}{2}) + 3 + 2 cos (θ) = 0

仮定:

u = \frac{θ}{2}

4 cos (u) + 3 + 2 cos (2 u) = 0

三角関数の公式を使用して書き換える

3 + 2 cos (2 u) + 4 cos (u)

2倍角の公式を使用:

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

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

簡素化

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

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

簡素化

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

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

数を乗じる:

2 \cdot 2 = 4

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

数を乗じる:

2 \cdot 1 = 2

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

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

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

簡素化

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

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

条件のようなグループ

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

数を足す/引く:

3 - 2 = 1

= 4 cos^{2} (u) + 4 cos (u) + 1

= 4 cos^{2} (u) + 4 cos (u) + 1

= 4 cos^{2} (u) + 4 cos (u) + 1

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

置換で解く

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

仮定:

cos (u) = 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

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}

数を乗じる:

2 \cdot 4 = 8

= \frac{- 4}{8}

分数の規則を適用する:

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

= - \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 (u)

cos (u) = - \frac{1}{2}

cos (u) = - \frac{1}{2}

cos (u) = - \frac{1}{2} : u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

cos (u) = - \frac{1}{2}

以下の一般解

cos (u) = - \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}

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

すべての解を組み合わせる

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

代用を戻す

u = \frac{θ}{2}

\frac{θ}{2} = \frac{2 π}{3} + 2 πn : θ = \frac{4 π}{3} + 4 πn

\frac{θ}{2} = \frac{2 π}{3} + 2 πn

以下で両辺を乗じる:

2

\frac{θ}{2} = \frac{2 π}{3} + 2 πn

以下で両辺を乗じる:

2

\frac{2 θ}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

簡素化

\frac{2 θ}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

簡素化

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数を割る:

\frac{2}{2} = 1

= θ

簡素化

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn : \frac{4 π}{3} + 4 πn

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 π 2}{3}

数を乗じる:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn

θ = \frac{4 π}{3} + 4 πn

θ = \frac{4 π}{3} + 4 πn

θ = \frac{4 π}{3} + 4 πn

\frac{θ}{2} = \frac{4 π}{3} + 2 πn : θ = \frac{8 π}{3} + 4 πn

\frac{θ}{2} = \frac{4 π}{3} + 2 πn

以下で両辺を乗じる:

2

\frac{θ}{2} = \frac{4 π}{3} + 2 πn

以下で両辺を乗じる:

2

\frac{2 θ}{2} = 2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

簡素化

\frac{2 θ}{2} = 2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

簡素化

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数を割る:

\frac{2}{2} = 1

= θ

簡素化

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn : \frac{8 π}{3} + 4 πn

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{4 π}{3} = \frac{8 π}{3}

2 \cdot \frac{4 π}{3}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 π 2}{3}

数を乗じる:

4 \cdot 2 = 8

= \frac{8 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{8 π}{3} + 4 πn

θ = \frac{8 π}{3} + 4 πn

θ = \frac{8 π}{3} + 4 πn

θ = \frac{8 π}{3} + 4 πn

θ = \frac{4 π}{3} + 4 πn, θ = \frac{8 π}{3} + 4 πn

グラフ

人気の例

8 cos (2 x) - 1 = 2

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

2 cos (2 x) + 3 = 23

sin(x)cos(2x)= 1/2 (1+sin(x))

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

e^{s i n (x)} = 2

(sec(x))/(cot(x)tan(x))=sin(x)

\frac{sec ( x )}{cot ( x ) tan ( x )} = sin (x)