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人気のある三角関数 >

証明する tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x))

解

証明する

tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

解

真

解答ステップ

tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

左側を操作する

tan (x + \frac{π}{3})

三角関数の公式を使用して書き換える

tan (x + \frac{π}{3})

基本的な三角関数の公式を使用する:

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

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

角の和の公式を使用する:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

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

角の和の公式を使用する:

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

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

簡素化

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

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

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

sin (x) cos (\frac{π}{3}) + cos (x) sin (\frac{π}{3})

簡素化

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

次の自明恒等式を使用する:

cos (\frac{π}{3}) = \frac{1}{2}

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{1}{2}

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

簡素化

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

次の自明恒等式を使用する:

sin (\frac{π}{3}) = \frac{3}{2}

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{3}{2}

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

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

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

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

簡素化

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

次の自明恒等式を使用する:

cos (\frac{π}{3}) = \frac{1}{2}

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{1}{2}

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

簡素化

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

次の自明恒等式を使用する:

sin (\frac{π}{3}) = \frac{3}{2}

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{3}{2}

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

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

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

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

簡素化

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

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

乗じる

cos (x) \frac{1}{2} : \frac{cos ( x )}{2}

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

分数を乗じる:

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

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

= \frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{c o s ( x )}{2} - \frac{3}{2} sin ( x )}

乗じる

\frac{3}{2} sin (x) : \frac{3 sin ( x )}{2}

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

分数を乗じる:

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

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

= \frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{c o s ( x )}{2} - \frac{3 s i n ( x )}{2}}

乗じる

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

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

分数を乗じる:

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

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

= \frac{\frac{3 c o s ( x )}{2} + \frac{1}{2} sin ( x )}{\frac{c o s ( x )}{2} - \frac{3 s i n ( x )}{2}}

乗じる

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

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

分数を乗じる:

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

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

= \frac{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}{\frac{c o s ( x )}{2} - \frac{3 s i n ( x )}{2}}

分数を組み合わせる

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

規則を適用

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

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

= \frac{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}{\frac{c o s ( x ) - 3 s i n ( x )}{2}}

分数を組み合わせる

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

規則を適用

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

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

= \frac{\frac{3 c o s ( x ) + s i n ( x )}{2}}{\frac{c o s ( x ) - 3 s i n ( x )}{2}}

分数を割る:

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

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

共通因数を約分する:

2

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

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

右側を操作する

\frac{tan ( x ) + 3}{1 - 3 tan ( x )}

サイン, コサインで表わす

\frac{3 + tan ( x )}{1 - 3 tan ( x )}

基本的な三角関数の公式を使用する:

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

= \frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - 3 \frac{s i n ( x )}{c o s ( x )}}

簡素化

\frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - 3 \frac{s i n ( x )}{c o s ( x )}} : \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

\frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - 3 \frac{s i n ( x )}{c o s ( x )}}

乗じる

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

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

分数を乗じる:

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

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

= \frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - \frac{3 s i n ( x )}{c o s ( x )}}

結合

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

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

元を分数に変換する:

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

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

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{3 cos ( x ) + sin ( x )}{cos ( x )}

= \frac{\frac{3 c o s ( x ) + s i n ( x )}{c o s ( x )}}{1 - \frac{3 s i n ( x )}{c o s ( x )}}

分数の規則を適用する:

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

= \frac{3 cos ( x ) + sin ( x )}{cos ( x ) ( 1 - \frac{s i n ( x ) 3}{c o s ( x )} )}

結合

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

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

元を分数に変換する:

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

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

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 ( x ) - sin ( x ) 3}{cos ( x )}

乗算:

1 \cdot cos (x) = cos (x)

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

= \frac{3 cos ( x ) + sin ( x )}{\frac{c o s ( x ) - 3 s i n ( x )}{c o s ( x )} cos ( x )}

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

= cos (x) - sin (x) 3

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

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

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

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

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p ro v e \frac{1 - 2 sin ^{2} ( x )}{1 - sin ^{2} ( x )} = 1 - tan (2 x)

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p ro v e sin (x + y) - sin (x - y) = 2 sin (y) cos (x)

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p ro v e sin (2 a) = 2 cos (a) sin (a)

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p ro v e sin (2 α) = \frac{2 tan ( α )}{1 + tan ^{2} ( α )}

証明する cot(x)sec(x)csc^2(x)-cot^3(x)sec(x)=csc(x)

p ro v e cot (x) sec (x) csc^{2} (x) - cot^{3} (x) sec (x) = csc (x)