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受欢迎的三角函数 >

证明 sin(2α)=(2tan(α))/(1+tan^2(α))

解答

证明

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

解答

真

求解步骤

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

调整右侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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}}

化简

\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}}

使用指数法则:

(\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 )}}

乘

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

2 \cdot \frac{sin ( α )}{cos ( α )}

分式相乘:

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 )}}

使用分式法则:

\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 )} )}

化简

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 )}

将项转换为分式:

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 )}

因为分母相等，所以合并分式:

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

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

乘以:

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 )}

乘

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 )}

分式相乘:

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

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

约分:

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 )}}

使用分式法则:

\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 )}

使用三角恒等式改写

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

使用倍角公式:

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

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

使用毕达哥拉斯恒等式:

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

= \frac{sin ( 2 α )}{1}

使用法则

\frac{a}{1} = a

= sin (2 α)

= sin (2 α)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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