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人気のある三角関数 >

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

解

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

解

x = \frac{πn}{2} + \frac{π}{16} + \frac{0.36136 \dots}{4}, x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{0.36136 \dots}{4}

+1

度

x = 16.42620 \dots^{\circ} + 9 0^{\circ} n, x = 51.07379 \dots^{\circ} + 9 0^{\circ} n

解答ステップ

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

三角関数の公式を使用して書き換える

sin (4 x) - cos (4 x)

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

sin (4 x) - cos (4 x)

書き換え

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

次の自明恒等式を使用する:

cos (\frac{π}{4}) = \frac{1}{2}

次の自明恒等式を使用する:

sin (\frac{π}{4}) = \frac{1}{2}

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

角の和の公式を使用する:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (4 x - \frac{π}{4})

= 2 sin (4 x - \frac{π}{4})

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

以下で両辺を割る

2

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

以下で両辺を割る

2

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

簡素化

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

簡素化

\frac{2 sin ( 4 x - \frac{π}{4} )}{2} : sin (4 x - \frac{π}{4})

\frac{2 sin ( 4 x - \frac{π}{4} )}{2}

共通因数を約分する:

2

= sin (4 x - \frac{π}{4})

簡素化

\frac{\frac{1}{2}}{2} : \frac{2}{4}

\frac{\frac{1}{2}}{2}

分数の規則を適用する:

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

= \frac{1}{2 2}

有理化する

\frac{1}{2 2} : \frac{2}{4}

\frac{1}{2 2}

共役で乗じる

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

指数の規則を適用する:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

類似した元を足す:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 2}{2}

共通因数を約分する:

2

= 1

= 2^{1 + 1}

数を足す:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4}

sin (4 x - \frac{π}{4}) = \frac{2}{4}

sin (4 x - \frac{π}{4}) = \frac{2}{4}

sin (4 x - \frac{π}{4}) = \frac{2}{4}

三角関数の逆数プロパティを適用する

sin (4 x - \frac{π}{4}) = \frac{2}{4}

以下の一般解

sin (4 x - \frac{π}{4}) = \frac{2}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

4 x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, 4 x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

4 x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, 4 x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

解く

4 x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn : x = \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}

4 x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn

簡素化

arcsin (\frac{2}{4}) + 2 πn : arcsin (\frac{1}{2 2}) + 2 πn

arcsin (\frac{2}{4}) + 2 πn

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

因数

4 : 2^{2}

因数

4 = 2^{2}

= \frac{2}{2 ^{2}}

キャンセル

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

指数の規則を適用する:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

数を引く:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

指数の規則を適用する:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

改良

= 22

= \frac{1}{2 2}

= arcsin (\frac{1}{2 2}) + 2 πn

4 x - \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn

\frac{π}{4}

を右側に移動します

4 x - \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn

両辺に

\frac{π}{4}

を足す

4 x - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

簡素化

4 x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

4 x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

以下で両辺を割る

4

4 x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

以下で両辺を割る

4

\frac{4 x}{4} = \frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

簡素化

\frac{4 x}{4} = \frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

簡素化

\frac{4 x}{4} : x

\frac{4 x}{4}

数を割る:

\frac{4}{4} = 1

= x

簡素化

\frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} : \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}

\frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

条件のようなグループ

= \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} + \frac{arcsin ( \frac{1}{2 2} )}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

共通因数を約分する:

2

= \frac{πn}{2}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

分数の規則を適用する:

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

= \frac{π}{4 \cdot 4}

数を乗じる:

4 \cdot 4 = 16

= \frac{π}{16}

= \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{1}{2 2} )}{4}

arcsin (\frac{1}{2 2}) = arcsin (\frac{2}{4})

arcsin (\frac{1}{2 2})

= arcsin (\frac{2}{4})

= \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}

解く

4 x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn : x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

4 x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

簡素化

π - arcsin (\frac{2}{4}) + 2 πn : π - arcsin (\frac{1}{2 2}) + 2 πn

π - arcsin (\frac{2}{4}) + 2 πn

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

因数

4 : 2^{2}

因数

4 = 2^{2}

= \frac{2}{2 ^{2}}

キャンセル

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

指数の規則を適用する:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

数を引く:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

指数の規則を適用する:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

改良

= 22

= \frac{1}{2 2}

= π - arcsin (\frac{1}{2 2}) + 2 πn

4 x - \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn

\frac{π}{4}

を右側に移動します

4 x - \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn

両辺に

\frac{π}{4}

を足す

4 x - \frac{π}{4} + \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

簡素化

4 x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

4 x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

以下で両辺を割る

4

4 x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

以下で両辺を割る

4

\frac{4 x}{4} = \frac{π}{4} - \frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

簡素化

\frac{4 x}{4} = \frac{π}{4} - \frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

簡素化

\frac{4 x}{4} : x

\frac{4 x}{4}

数を割る:

\frac{4}{4} = 1

= x

簡素化

\frac{π}{4} - \frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} : \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

\frac{π}{4} - \frac{arcsin ( \frac{1}{2 2} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

条件のようなグループ

= \frac{π}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} - \frac{arcsin ( \frac{1}{2 2} )}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

共通因数を約分する:

2

= \frac{πn}{2}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

分数の規則を適用する:

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

= \frac{π}{4 \cdot 4}

数を乗じる:

4 \cdot 4 = 16

= \frac{π}{16}

= \frac{π}{4} + \frac{πn}{2} + \frac{π}{16} - \frac{arcsin ( \frac{1}{2 2} )}{4}

条件のようなグループ

= \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{1}{2 2} )}{4}

= \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

因数

4 : 2^{2}

因数

4 = 2^{2}

= \frac{2}{2 ^{2}}

キャンセル

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

指数の規則を適用する:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

数を引く:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

指数の規則を適用する:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

改良

= 22

= \frac{1}{2 2}

= \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{1}{2 2} )}{4}

arcsin (\frac{1}{2 2}) = arcsin (\frac{2}{4})

arcsin (\frac{1}{2 2})

= arcsin (\frac{2}{4})

= \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

x = \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( \frac{2}{4} )}{4}, x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{arcsin ( \frac{2}{4} )}{4}

10進法形式で解を証明する

x = \frac{πn}{2} + \frac{π}{16} + \frac{0.36136 \dots}{4}, x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} - \frac{0.36136 \dots}{4}

グラフ

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