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人気のある三角関数 >

4sin(x/2)+4cos(x)=0

解

4 sin (\frac{x}{2}) + 4 cos (x) = 0

解

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = π + 4 πn

+1

度

x = 42 0^{\circ} + 72 0^{\circ} n, x = 66 0^{\circ} + 72 0^{\circ} n, x = 18 0^{\circ} + 72 0^{\circ} n

解答ステップ

4 sin (\frac{x}{2}) + 4 cos (x) = 0

仮定:

u = \frac{x}{2}

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

三角関数の公式を使用して書き換える

4 cos (2 u) + 4 sin (u)

2倍角の公式を使用:

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

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

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

置換で解く

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

仮定:

sin (u) = u

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

(1 - 2 u^{2}) \cdot 4 + 4 u = 0 : u = - \frac{1}{2}, u = 1

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

拡張

(1 - 2 u^{2}) \cdot 4 + 4 u : 4 - 8 u^{2} + 4 u

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

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

拡張

4 (1 - 2 u^{2}) : 4 - 8 u^{2}

4 (1 - 2 u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 4, b = 1, c = 2 u^{2}

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

簡素化

4 \cdot 1 - 4 \cdot 2 u^{2} : 4 - 8 u^{2}

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

数を乗じる:

4 \cdot 1 = 4

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

数を乗じる:

4 \cdot 2 = 8

= 4 - 8 u^{2}

= 4 - 8 u^{2}

= 4 - 8 u^{2} + 4 u

4 - 8 u^{2} + 4 u = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 8, b = 4, c = 4

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 8 ) \cdot 4}{2 ( - 8 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 8 ) \cdot 4}{2 ( - 8 )}

4^{2} - 4 (- 8) \cdot 4 = 12

4^{2} - 4 (- 8) \cdot 4

規則を適用

- (- a) = a

= 4^{2} + 4 \cdot 8 \cdot 4

数を乗じる:

4 \cdot 8 \cdot 4 = 128

= 4^{2} + 128

4^{2} = 16

= 16 + 128

数を足す:

16 + 128 = 144

= 144

数を因数に分解する:

144 = 1 2^{2}

= 1 2^{2}

累乗根の規則を適用する:

n a^{n} = a

1 2^{2} = 12

= 12

u_{1, 2} = \frac{- 4 \pm 12}{2 ( - 8 )}

解を分離する

u_{1} = \frac{- 4 + 12}{2 ( - 8 )}, u_{2} = \frac{- 4 - 12}{2 ( - 8 )}

u = \frac{- 4 + 12}{2 ( - 8 )} : - \frac{1}{2}

\frac{- 4 + 12}{2 ( - 8 )}

括弧を削除する:

(- a) = - a

= \frac{- 4 + 12}{- 2 \cdot 8}

数を足す/引く:

- 4 + 12 = 8

= \frac{8}{- 2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{8}{- 16}

分数の規則を適用する:

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

= - \frac{8}{16}

共通因数を約分する:

8

= - \frac{1}{2}

u = \frac{- 4 - 12}{2 ( - 8 )} : 1

\frac{- 4 - 12}{2 ( - 8 )}

括弧を削除する:

(- a) = - a

= \frac{- 4 - 12}{- 2 \cdot 8}

数を引く:

- 4 - 12 = - 16

= \frac{- 16}{- 2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{- 16}{- 16}

分数の規則を適用する:

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

= \frac{16}{16}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

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

代用を戻す

u = sin (u)

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

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

sin (u) = - \frac{1}{2} : u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

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

以下の一般解

sin (u) = - \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}

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

sin (u) = 1 : u = \frac{π}{2} + 2 πn

sin (u) = 1

以下の一般解

sin (u) = 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}

u = \frac{π}{2} + 2 πn

u = \frac{π}{2} + 2 πn

すべての解を組み合わせる

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn, u = \frac{π}{2} + 2 πn

代用を戻す

u = \frac{x}{2}

\frac{x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{7 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

分数を乗じる:

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

= \frac{7 π 2}{6}

数を乗じる:

7 \cdot 2 = 14

= \frac{14 π}{6}

共通因数を約分する:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{11 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

分数を乗じる:

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

= \frac{11 π 2}{6}

数を乗じる:

11 \cdot 2 = 22

= \frac{22 π}{6}

共通因数を約分する:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{π}{2} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分数を乗じる:

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

= \frac{π 2}{2}

共通因数を約分する:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = π + 4 πn

グラフ

人気の例

cos(θ)cos(θ)=cot(θ)

cos (θ) cos (θ) = cot (θ)

3sin(θ)-cos(2θ)=1

3 sin (θ) - cos (2 θ) = 1

sin(θ)= 28/53 ,sin(2θ)

sin (θ) = \frac{28}{53}, sin (2 θ)

arctan (x) = 5^{\circ}

tan (t) = 5