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人気のある三角関数 >

tan(2a)cot(a+20)=1

解

tan (2 a) cot (a + 2 0^{\circ}) = 1

解

a = 36 0^{\circ} n + 2 0^{\circ}, a = 20 0^{\circ} + 36 0^{\circ} n

+1

ラジアン

a = \frac{π}{9} + 2 πn, a = \frac{10 π}{9} + 2 πn

解答ステップ

tan (2 a) cot (a + 2 0^{\circ}) = 1

両辺から

1

を引く

tan (2 a) cot (a + 2 0^{\circ}) - 1 = 0

サイン, コサインで表わす

- 1 + cot (2 0^{\circ} + a) tan (2 a)

基本的な三角関数の公式を使用する:

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

= - 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} tan (2 a)

基本的な三角関数の公式を使用する:

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

= - 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )}

簡素化

- 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )} : \frac{- sin ( \frac{18 0 ^{\circ} + 9 a}{9} ) cos ( 2 a ) + cos ( \frac{18 0 ^{\circ} + 9 a}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + 9 a}{9} ) cos ( 2 a )}

- 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )}

\frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )} = \frac{cos ( \frac{18 0 ^{\circ} + 9 a}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + 9 a}{9} ) cos ( 2 a )}

\frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )}

分数を乗じる:

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

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

結合

2 0^{\circ} + a : \frac{18 0 ^{\circ} + 9 a}{9}

2 0^{\circ} + a

元を分数に変換する:

a = \frac{a 9}{9}

= 2 0^{\circ} + \frac{a \cdot 9}{9}

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{18 0 ^{\circ} + a \cdot 9}{9}

= \frac{cos ( a + 2 0 ^{\circ} ) sin ( 2 a )}{sin ( \frac{9 a + 18 0 ^{\circ}}{9} ) cos ( 2 a )}

結合

2 0^{\circ} + a : \frac{18 0 ^{\circ} + 9 a}{9}

2 0^{\circ} + a

元を分数に変換する:

a = \frac{a 9}{9}

= 2 0^{\circ} + \frac{a \cdot 9}{9}

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{18 0 ^{\circ} + a \cdot 9}{9}

= \frac{cos ( \frac{9 a + 18 0 ^{\circ}}{9} ) sin ( 2 a )}{sin ( \frac{9 a + 18 0 ^{\circ}}{9} ) cos ( 2 a )}

= - 1 + \frac{cos ( \frac{9 a + 18 0 ^{\circ}}{9} ) sin ( 2 a )}{sin ( \frac{9 a + 18 0 ^{\circ}}{9} ) cos ( 2 a )}

元を分数に変換する:

1 = \frac{1 sin ( \frac{18 0 ^{\circ} + a 9}{9} ) cos ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a 9}{9} ) cos ( 2 a )}

= - \frac{1 \cdot sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )} + \frac{cos ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )}

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 sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a ) + cos ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )}

乗算:

1 \cdot sin (\frac{18 0 ^{\circ} + a \cdot 9}{9}) = sin (\frac{18 0 ^{\circ} + a \cdot 9}{9})

= \frac{- sin ( \frac{9 a + 18 0 ^{\circ}}{9} ) cos ( 2 a ) + cos ( \frac{9 a + 18 0 ^{\circ}}{9} ) sin ( 2 a )}{sin ( \frac{9 a + 18 0 ^{\circ}}{9} ) cos ( 2 a )}

= \frac{- sin ( \frac{18 0 ^{\circ} + 9 a}{9} ) cos ( 2 a ) + cos ( \frac{18 0 ^{\circ} + 9 a}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + 9 a}{9} ) cos ( 2 a )}

\frac{- cos ( 2 a ) sin ( \frac{18 0 ^{\circ} + 9 a}{9} ) + cos ( \frac{18 0 ^{\circ} + 9 a}{9} ) sin ( 2 a )}{cos ( 2 a ) sin ( \frac{18 0 ^{\circ} + 9 a}{9} )} = 0

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

- cos (2 a) sin (\frac{18 0 ^{\circ} + 9 a}{9}) + cos (\frac{18 0 ^{\circ} + 9 a}{9}) sin (2 a) = 0

三角関数の公式を使用して書き換える

- cos (2 a) sin (\frac{18 0 ^{\circ} + 9 a}{9}) + cos (\frac{18 0 ^{\circ} + 9 a}{9}) sin (2 a)

角の差の公式を使用する:

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

= sin (2 a - \frac{18 0 ^{\circ} + 9 a}{9})

sin (2 a - \frac{18 0 ^{\circ} + 9 a}{9}) = 0

以下の一般解

sin (2 a - \frac{18 0 ^{\circ} + 9 a}{9}) = 0

sin (x)

36 0^{\circ} n

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}

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n, 2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n, 2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n

解く

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n : a = 36 0^{\circ} n + 2 0^{\circ}

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 36 0^{\circ} n

以下で両辺を乗じる:

9

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 36 0^{\circ} n

以下で両辺を乗じる:

9

2 a \cdot 9 - \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 = 36 0^{\circ} n \cdot 9

簡素化

2 a \cdot 9 - \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 = 36 0^{\circ} n \cdot 9

簡素化

2 a \cdot 9 : 18 a

2 a \cdot 9

数を乗じる:

2 \cdot 9 = 18

= 18 a

簡素化

- \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 : - (18 0^{\circ} + 9 a)

- \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9

分数を乗じる:

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

= - \frac{( 18 0 ^{\circ} + 9 a ) \cdot 9}{9}

共通因数を約分する:

9

= - (9 a + 18 0^{\circ})

簡素化

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

数を乗じる:

2 \cdot 9 = 18

= 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 324 0^{\circ} n

拡張

18 a - (18 0^{\circ} + 9 a) : 9 a - 18 0^{\circ}

18 a - (18 0^{\circ} + 9 a)

- (18 0^{\circ} + 9 a) : - 18 0^{\circ} - 9 a

- (18 0^{\circ} + 9 a)

括弧を分配する

= - (18 0^{\circ}) - (9 a)

マイナス・プラスの規則を適用する

+ (- a) = - a

= - 18 0^{\circ} - 9 a

= 18 a - 18 0^{\circ} - 9 a

簡素化

18 a - 18 0^{\circ} - 9 a : 9 a - 18 0^{\circ}

18 a - 18 0^{\circ} - 9 a

条件のようなグループ

= 18 a - 9 a - 18 0^{\circ}

類似した元を足す:

18 a - 9 a = 9 a

= 9 a - 18 0^{\circ}

= 9 a - 18 0^{\circ}

9 a - 18 0^{\circ} = 324 0^{\circ} n

18 0^{\circ}

を右側に移動します

9 a - 18 0^{\circ} = 324 0^{\circ} n

両辺に

18 0^{\circ}

を足す

9 a - 18 0^{\circ} + 18 0^{\circ} = 324 0^{\circ} n + 18 0^{\circ}

簡素化

9 a = 324 0^{\circ} n + 18 0^{\circ}

9 a = 324 0^{\circ} n + 18 0^{\circ}

以下で両辺を割る

9

9 a = 324 0^{\circ} n + 18 0^{\circ}

以下で両辺を割る

9

\frac{9 a}{9} = \frac{324 0 ^{\circ} n}{9} + 2 0^{\circ}

簡素化

a = 36 0^{\circ} n + 2 0^{\circ}

a = 36 0^{\circ} n + 2 0^{\circ}

解く

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n : a = 20 0^{\circ} + 36 0^{\circ} n

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n

以下で両辺を乗じる:

9

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n

以下で両辺を乗じる:

9

2 a \cdot 9 - \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 = 18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9

簡素化

2 a \cdot 9 - \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 = 18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9

簡素化

2 a \cdot 9 : 18 a

2 a \cdot 9

数を乗じる:

2 \cdot 9 = 18

= 18 a

簡素化

- \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 : - (18 0^{\circ} + 9 a)

- \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9

分数を乗じる:

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

= - \frac{( 18 0 ^{\circ} + 9 a ) \cdot 9}{9}

共通因数を約分する:

9

= - (9 a + 18 0^{\circ})

簡素化

18 0^{\circ} 9 : 162 0^{\circ}

18 0^{\circ} 9

交換法則を適用する:

18 0^{\circ} 9 = 162 0^{\circ}

162 0^{\circ}

簡素化

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

数を乗じる:

2 \cdot 9 = 18

= 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 162 0^{\circ} + 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 162 0^{\circ} + 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 162 0^{\circ} + 324 0^{\circ} n

拡張

18 a - (18 0^{\circ} + 9 a) : 9 a - 18 0^{\circ}

18 a - (18 0^{\circ} + 9 a)

- (18 0^{\circ} + 9 a) : - 18 0^{\circ} - 9 a

- (18 0^{\circ} + 9 a)

括弧を分配する

= - (18 0^{\circ}) - (9 a)

マイナス・プラスの規則を適用する

+ (- a) = - a

= - 18 0^{\circ} - 9 a

= 18 a - 18 0^{\circ} - 9 a

簡素化

18 a - 18 0^{\circ} - 9 a : 9 a - 18 0^{\circ}

18 a - 18 0^{\circ} - 9 a

条件のようなグループ

= 18 a - 9 a - 18 0^{\circ}

類似した元を足す:

18 a - 9 a = 9 a

= 9 a - 18 0^{\circ}

= 9 a - 18 0^{\circ}

9 a - 18 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n

18 0^{\circ}

を右側に移動します

9 a - 18 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n

両辺に

18 0^{\circ}

を足す

9 a - 18 0^{\circ} + 18 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n + 18 0^{\circ}

簡素化

9 a = 180 0^{\circ} + 324 0^{\circ} n

9 a = 180 0^{\circ} + 324 0^{\circ} n

以下で両辺を割る

9

9 a = 180 0^{\circ} + 324 0^{\circ} n

以下で両辺を割る

9

\frac{9 a}{9} = 20 0^{\circ} + \frac{324 0 ^{\circ} n}{9}

簡素化

a = 20 0^{\circ} + 36 0^{\circ} n

a = 20 0^{\circ} + 36 0^{\circ} n

a = 36 0^{\circ} n + 2 0^{\circ}, a = 20 0^{\circ} + 36 0^{\circ} n

グラフ

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