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人気のある >

2sin^2(1/2 x)+C=-2cos(x/2)sin(x/2)+C

解

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

解

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

+1

度

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

- 2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

を引く

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

因数

2 sin^{2} (\frac{x}{2}) + 2 cos (\frac{x}{2}) sin (\frac{x}{2}) : 2 sin (\frac{x}{2}) (sin (\frac{x}{2}) + cos (\frac{x}{2}))

2 sin^{2} (\frac{x}{2}) + 2 cos (\frac{x}{2}) sin (\frac{x}{2})

指数の規則を適用する:

a^{b + c} = a^{b} a^{c}

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

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

共通項をくくり出す

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

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

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

各部分を別個に解く

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

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

sin (\frac{x}{2}) = 0

以下の一般解

sin (\frac{x}{2}) = 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}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

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

解く

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot 2 πn

簡素化

x = 4 πn

x = 4 πn

解く

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

\frac{x}{2} = π + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = π + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

簡素化

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) + cos (\frac{x}{2}) = 0 : x = \frac{3 π}{2} + 2 πn

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

三角関数の公式を使用して書き換える

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

cos (\frac{x}{2}), cos (\frac{x}{2}) \neq = 0

で両辺を割る

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

簡素化

\frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )} + 1 = 0

基本的な三角関数の公式を使用する:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (\frac{x}{2}) + 1 = 0

tan (\frac{x}{2}) + 1 = 0

1

を右側に移動します

tan (\frac{x}{2}) + 1 = 0

両辺から

1

を引く

tan (\frac{x}{2}) + 1 - 1 = 0 - 1

簡素化

tan (\frac{x}{2}) = - 1

tan (\frac{x}{2}) = - 1

以下の一般解

tan (\frac{x}{2}) = - 1

tan (x)

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{3 π}{4} + πn

\frac{x}{2} = \frac{3 π}{4} + πn

解く

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

\frac{x}{2} = \frac{3 π}{4} + πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{3 π}{4} + πn

以下で両辺を乗じる:

2

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

簡素化

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

2 \cdot \frac{3 π}{4} + 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

分数を乗じる:

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

= \frac{3 π 2}{4}

数を乗じる:

3 \cdot 2 = 6

= \frac{6 π}{4}

共通因数を約分する:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

すべての解を組み合わせる

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

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