ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある三角関数 >

証明する sin(2α)=(2tan(α))/(1+tan^2(α))

解

証明する

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

解

真

解答ステップ

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

右側を操作する

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

サイン, コサインで表わす

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

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

簡素化

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

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

乗じる

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

2 \cdot \frac{sin ( α )}{cos ( α )}

分数を乗じる:

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

= \frac{sin ( α ) \cdot 2}{cos ( α )}

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

分数の規則を適用する:

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

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

結合

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

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

元を分数に変換する:

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

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

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} ( α ) + sin ^{2} ( α )}{cos ^{2} ( α )}

乗算:

1 \cdot cos^{2} (α) = cos^{2} (α)

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

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

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (α)

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

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

分数の規則を適用する:

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

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

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

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

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

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

ピタゴラスの公式を使用する:

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

= \frac{sin ( 2 α )}{1}

規則を適用

\frac{a}{1} = a

= sin (2 α)

= sin (2 α)

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 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)

証明する csc(t)-sin(t)=cot(t)cos(t)

p ro v e csc (t) - sin (t) = cot (t) cos (t)

証明する cos(60)=2cos^2(30)-1

p ro v e cos (6 0^{\circ}) = 2 cos^{2} (3 0^{\circ}) - 1

証明する (sin(t)+cos(t))^2=1+sin(2t)

p ro v e (sin (t) + cos (t))^{2} = 1 + sin (2 t)

証明する cos^2(x/2)=(sec(x)+1)/(2sec(x))

p ro v e cos^{2} (\frac{x}{2}) = \frac{sec ( x ) + 1}{2 sec ( x )}