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人気のある三角関数 >

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

解

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

解

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

+1

度

x = 9 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

解答ステップ

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

両辺から

1

を引く

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

サイン, コサインで表わす

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

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

元を分数に変換する:

1 = \frac{1 cos ( \frac{x}{2} )}{cos ( \frac{x}{2} )}, cos (x) = \frac{cos ( x ) cos ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

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

分母が等しいので, 分数を組み合わせる:

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

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

乗算:

1 \cdot cos (\frac{x}{2}) = cos (\frac{x}{2})

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

三角関数の公式を使用して書き換える

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

積・和の公式を使用する:

cos (s) cos (t) = \frac{1}{2} (cos (s - t) + cos (s + t))

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

簡素化

- cos (\frac{x}{2}) + sin (\frac{x}{2}) + \frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)) : \frac{- cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) + 2 sin ( \frac{x}{2} )}{2}

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

\frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)) = \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

\frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

分数を乗じる:

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

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

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

1 \cdot (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

乗算:

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

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

括弧を削除する:

(a) = a

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

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

結合

\frac{x}{2} - x : - \frac{x}{2}

\frac{x}{2} - x

元を分数に変換する:

x = \frac{x 2}{2}

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

分母が等しいので, 分数を組み合わせる:

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

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

類似した元を足す:

x - 2 x = - x

= \frac{- x}{2}

分数の規則を適用する:

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

= - \frac{x}{2}

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

結合

\frac{x}{2} + x : \frac{3 x}{2}

\frac{x}{2} + x

元を分数に変換する:

x = \frac{x 2}{2}

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

分母が等しいので, 分数を組み合わせる:

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

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

類似した元を足す:

x + 2 x = 3 x

= \frac{3 x}{2}

= \frac{cos ( - \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

簡素化

cos (- \frac{x}{2}) + cos (\frac{3 x}{2}) : cos (\frac{x}{2}) + cos (\frac{3 x}{2})

cos (- \frac{x}{2}) + cos (\frac{3 x}{2})

負角の公式を使用する:

cos (- x) = cos (x)

= cos (\frac{x}{2}) + cos (\frac{3 x}{2})

= \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

= - cos (\frac{x}{2}) + sin (\frac{x}{2}) + \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

元を分数に変換する:

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

= \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2} - \frac{cos ( \frac{x}{2} ) \cdot 2}{2} + \frac{sin ( \frac{x}{2} ) \cdot 2}{2}

分母が等しいので, 分数を組み合わせる:

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

= \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) - cos ( \frac{x}{2} ) \cdot 2 + sin ( \frac{x}{2} ) \cdot 2}{2}

cos (\frac{x}{2}) + cos (\frac{3 x}{2}) - cos (\frac{x}{2}) \cdot 2 + sin (\frac{x}{2}) \cdot 2 = - cos (\frac{x}{2}) + cos (\frac{3 x}{2}) + 2 sin (\frac{x}{2})

cos (\frac{x}{2}) + cos (\frac{3 x}{2}) - cos (\frac{x}{2}) \cdot 2 + sin (\frac{x}{2}) \cdot 2

条件のようなグループ

= cos (\frac{x}{2}) + cos (\frac{3 x}{2}) - 2 cos (\frac{x}{2}) + 2 sin (\frac{x}{2})

類似した元を足す:

cos (\frac{x}{2}) - 2 cos (\frac{x}{2}) = - cos (\frac{x}{2})

= - cos (\frac{x}{2}) + cos (\frac{3 x}{2}) + 2 sin (\frac{x}{2})

= \frac{- cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) + 2 sin ( \frac{x}{2} )}{2}

= \frac{- cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) + 2 sin ( \frac{x}{2} )}{2}

和・積の公式を使用する:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= \frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{\frac{3 x}{2} - \frac{x}{2}}{2} )}{2}

簡素化

\frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{\frac{3 x}{2} - \frac{x}{2}}{2} )}{2} : sin (\frac{x}{2}) (- sin (x) + 1)

\frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{\frac{3 x}{2} - \frac{x}{2}}{2} )}{2}

分数を組み合わせる

\frac{3 x}{2} - \frac{x}{2} : x

規則を適用

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

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

類似した元を足す:

3 x - x = 2 x

= \frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

= \frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{x}{2} )}{2}

分数を組み合わせる

\frac{3 x}{2} + \frac{x}{2} : 2 x

規則を適用

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

= \frac{3 x + x}{2}

類似した元を足す:

3 x + x = 4 x

= \frac{4 x}{2}

数を割る:

\frac{4}{2} = 2

= 2 x

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

因数

2 sin (\frac{x}{2}) - 2 sin (\frac{2 x}{2}) sin (\frac{x}{2}) : 2 sin (\frac{x}{2}) (1 - sin (x))

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

書き換え

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

共通項をくくり出す

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

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

改良

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

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

数を割る:

\frac{2}{2} = 1

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

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

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

各部分を別個に解く

1 - sin (x) = 0 or sin (\frac{x}{2}) = 0

1 - sin (x) = 0 : x = \frac{π}{2} + 2 πn

1 - sin (x) = 0

1

を右側に移動します

1 - sin (x) = 0

両辺から

1

を引く

1 - sin (x) - 1 = 0 - 1

簡素化

- sin (x) = - 1

- sin (x) = - 1

以下で両辺を割る

- 1

- sin (x) = - 1

以下で両辺を割る

- 1

\frac{- sin ( x )}{- 1} = \frac{- 1}{- 1}

簡素化

sin (x) = 1

sin (x) = 1

以下の一般解

sin (x) = 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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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

すべての解を組み合わせる

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

グラフ

人気の例

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1 + cos^{2} (a) = 2 cos^{2} (a)

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cos (5 x) = cos (5 + x)

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