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人気のある三角関数 >

証明する sin(2a)=(2tan(a))/(1+tan^2(a))

解

証明する

sin (2 a) = \frac{2 tan ( a )}{1 + tan ^{2} ( a )}

解

真

解答ステップ

sin (2 a) = \frac{2 tan ( a )}{1 + tan ^{2} ( a )}

右側を操作する

\frac{2 tan ( a )}{1 + tan ^{2} ( a )}

サイン, コサインで表わす

\frac{2 tan ( a )}{1 + 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 )}

= \frac{2 \cdot \frac{s i n ( a )}{c o s ( a )}}{1 + ( \frac{s i n ( a )}{c o s ( a )} ) ^{2}}

簡素化

\frac{2 \cdot \frac{s i n ( a )}{c o s ( a )}}{1 + ( \frac{s i n ( a )}{c o s ( a )} ) ^{2}} : \frac{2 sin ( a ) cos ( a )}{cos ^{2} ( a ) + sin ^{2} ( a )}

\frac{2 \cdot \frac{s i n ( a )}{c o s ( a )}}{1 + ( \frac{s i n ( a )}{c o s ( a )} ) ^{2}}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{2 \cdot \frac{s i n ( a )}{c o s ( a )}}{1 + \frac{s i n ^{2} ( a )}{c o s ^{2} ( a )}}

乗じる

2 \cdot \frac{sin ( a )}{cos ( a )} : \frac{2 sin ( a )}{cos ( a )}

2 \cdot \frac{sin ( a )}{cos ( a )}

分数を乗じる:

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

= \frac{sin ( a ) \cdot 2}{cos ( a )}

= \frac{\frac{2 s i n ( a )}{c o s ( a )}}{1 + \frac{s i n ^{2} ( a )}{c o s ^{2} ( a )}}

分数の規則を適用する:

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

= \frac{sin ( a ) \cdot 2}{cos ( a ) ( 1 + \frac{s i n ^{2} ( a )}{c o s ^{2} ( a )} )}

結合

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

1 + \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

元を分数に変換する:

1 = \frac{1 cos ^{2} ( a )}{cos ^{2} ( a )}

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

乗算:

1 \cdot cos^{2} (a) = cos^{2} (a)

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

= \frac{2 sin ( a )}{\frac{c o s ^{2} ( a ) + s i n ^{2} ( a )}{c o s ^{2} ( a )} cos ( a )}

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (a)

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

= \frac{2 sin ( a )}{\frac{c o s ^{2} ( a ) + s i n ^{2} ( a )}{c o s ( a )}}

分数の規則を適用する:

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

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

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

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

2 sin (x) cos (x) = sin (2 x)

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

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

= \frac{sin ( 2 a )}{1}

規則を適用

\frac{a}{1} = a

= sin (2 a)

= sin (2 a)

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

証明する tan(x)=((1-cos(2x)))/(sin(2x))

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

証明する sin^2(a)-cos^2(a)=1-2cos^2(a)

p ro v e sin^{2} (a) - cos^{2} (a) = 1 - 2 cos^{2} (a)

証明する tan(2A)-2tan(2A)sin^2(A)=sin(2A)

p ro v e tan (2 A) - 2 tan (2 A) sin^{2} (A) = sin (2 A)

証明する sin^2(x)= 1/(csc^2(x))

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

証明する (cot(4u)-1)/(cot(4u)+1)=(1-tan(4u))/(1+tan(4u))

p ro v e \frac{cot ( 4 u ) - 1}{cot ( 4 u ) + 1} = \frac{1 - tan ( 4 u )}{1 + tan ( 4 u )}