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人気のある三角関数 >

cos(x/4)sin(x/4)=sqrt(3)sin(x/4)cos(x/4)

解

cos (\frac{x}{4}) sin (\frac{x}{4}) = 3 sin (\frac{x}{4}) cos (\frac{x}{4})

解

x = 2 π + 8 πn, x = 6 π + 8 πn, x = 8 πn, x = 4 π + 8 πn

+1

度

x = 36 0^{\circ} + 144 0^{\circ} n, x = 108 0^{\circ} + 144 0^{\circ} n, x = 0^{\circ} + 144 0^{\circ} n, x = 72 0^{\circ} + 144 0^{\circ} n

解答ステップ

cos (\frac{x}{4}) sin (\frac{x}{4}) = 3 sin (\frac{x}{4}) cos (\frac{x}{4})

両辺から

3 sin (\frac{x}{4}) cos (\frac{x}{4})

を引く

cos (\frac{x}{4}) sin (\frac{x}{4}) - 3 sin (\frac{x}{4}) cos (\frac{x}{4}) = 0

因数

cos (\frac{x}{4}) sin (\frac{x}{4}) - 3 sin (\frac{x}{4}) cos (\frac{x}{4}) : (1 - 3) cos (\frac{x}{4}) sin (\frac{x}{4})

cos (\frac{x}{4}) sin (\frac{x}{4}) - 3 sin (\frac{x}{4}) cos (\frac{x}{4})

共通項をくくり出す

cos (\frac{x}{4}) sin (\frac{x}{4})

= cos (\frac{x}{4}) sin (\frac{x}{4}) (1 - 3)

(1 - 3) cos (\frac{x}{4}) sin (\frac{x}{4}) = 0

各部分を別個に解く

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

cos (\frac{x}{4}) = 0 : x = 2 π + 8 πn, x = 6 π + 8 πn

cos (\frac{x}{4}) = 0

以下の一般解

cos (\frac{x}{4}) = 0

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}

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

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

解く

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

\frac{x}{4} = \frac{π}{2} + 2 πn

以下で両辺を乗じる:

4

\frac{x}{4} = \frac{π}{2} + 2 πn

以下で両辺を乗じる:

4

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

簡素化

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

簡素化

\frac{4 x}{4} : x

\frac{4 x}{4}

数を割る:

\frac{4}{4} = 1

= x

簡素化

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

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

分数を乗じる:

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

= \frac{π 4}{2}

数を割る:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数を乗じる:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

x = 2 π + 8 πn

x = 2 π + 8 πn

x = 2 π + 8 πn

解く

\frac{x}{4} = \frac{3 π}{2} + 2 πn : x = 6 π + 8 πn

\frac{x}{4} = \frac{3 π}{2} + 2 πn

以下で両辺を乗じる:

4

\frac{x}{4} = \frac{3 π}{2} + 2 πn

以下で両辺を乗じる:

4

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

簡素化

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

簡素化

\frac{4 x}{4} : x

\frac{4 x}{4}

数を割る:

\frac{4}{4} = 1

= x

簡素化

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

分数を乗じる:

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

= \frac{3 π 4}{2}

数を乗じる:

3 \cdot 4 = 12

= \frac{12 π}{2}

数を割る:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数を乗じる:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

x = 6 π + 8 πn

x = 6 π + 8 πn

x = 6 π + 8 πn

x = 2 π + 8 πn, x = 6 π + 8 πn

sin (\frac{x}{4}) = 0 : x = 8 πn, x = 4 π + 8 πn

sin (\frac{x}{4}) = 0

以下の一般解

sin (\frac{x}{4}) = 0

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}

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

解く

\frac{x}{4} = 0 + 2 πn : x = 8 πn

\frac{x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{4} = 2 πn

以下で両辺を乗じる:

4

\frac{x}{4} = 2 πn

以下で両辺を乗じる:

4

\frac{4 x}{4} = 4 \cdot 2 πn

簡素化

x = 8 πn

x = 8 πn

解く

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

\frac{x}{4} = π + 2 πn

以下で両辺を乗じる:

4

\frac{x}{4} = π + 2 πn

以下で両辺を乗じる:

4

\frac{4 x}{4} = 4 π + 4 \cdot 2 πn

簡素化

x = 4 π + 8 πn

x = 4 π + 8 πn

x = 8 πn, x = 4 π + 8 πn

すべての解を組み合わせる

x = 2 π + 8 πn, x = 6 π + 8 πn, x = 8 πn, x = 4 π + 8 πn

グラフ

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