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受欢迎的三角函数 >

证明 (cot(3θ)-tan(3θ))/(sin(3θ)+cos(3θ))=csc(3θ)-sec(3θ)

解答

证明

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )} = csc (3 θ) - sec (3 θ)

解答

真

求解步骤

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )} = csc (3 θ) - sec (3 θ)

调整左侧

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )}

用 sin, cos 表示

\frac{cot ( 3 θ ) - tan ( 3 θ )}{cos ( 3 θ ) + sin ( 3 θ )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

sin (3 θ), cos (3 θ)

的最小公倍数:

sin (3 θ) cos (3 θ)

sin (3 θ), cos (3 θ)

最小公倍数 (LCM)

计算出由出现在

sin (3 θ)

或

cos (3 θ)

中的因子组成的表达式

= sin (3 θ) cos (3 θ)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (3 θ) cos (3 θ)

对于

\frac{cos ( 3 θ )}{sin ( 3 θ )} :

将分母和分子乘以

cos (3 θ)

\frac{cos ( 3 θ )}{sin ( 3 θ )} = \frac{cos ( 3 θ ) cos ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )} = \frac{cos ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

对于

\frac{sin ( 3 θ )}{cos ( 3 θ )} :

将分母和分子乘以

sin (3 θ)

\frac{sin ( 3 θ )}{cos ( 3 θ )} = \frac{sin ( 3 θ ) sin ( 3 θ )}{cos ( 3 θ ) sin ( 3 θ )} = \frac{sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

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

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{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

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

使用分式法则:

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

= \frac{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ ) ( cos ( 3 θ ) + sin ( 3 θ ) )}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (3 θ) - sin^{2} (3 θ) = (cos (3 θ) + sin (3 θ)) (cos (3 θ) - sin (3 θ))

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

约分:

cos (3 θ) + sin (3 θ)

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

sin (x) = \frac{1}{csc ( x )}

\frac{cos ( 3 θ ) - \frac{1}{c s c ( 3 θ )}}{cos ( 3 θ ) \frac{1}{c s c ( 3 θ )}}

使用基本三角恒等式:

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

\frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ )} \cdot \frac{1}{c s c ( 3 θ )}}

化简

\frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ )} \cdot \frac{1}{c s c ( 3 θ )}}

乘

\frac{1}{sec ( 3 θ )} \cdot \frac{1}{csc ( 3 θ )} : \frac{1}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{sec ( 3 θ )} \cdot \frac{1}{csc ( 3 θ )}

分式相乘:

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

= \frac{1 \cdot 1}{sec ( 3 θ ) csc ( 3 θ )}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ ) c s c ( 3 θ )}}

化简

\frac{1}{sec ( 3 θ )} - \frac{1}{csc ( 3 θ )} : \frac{csc ( 3 θ ) - sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{sec ( 3 θ )} - \frac{1}{csc ( 3 θ )}

sec (3 θ), csc (3 θ)

的最小公倍数:

sec (3 θ) csc (3 θ)

sec (3 θ), csc (3 θ)

最小公倍数 (LCM)

计算出由出现在

sec (3 θ)

或

csc (3 θ)

中的因子组成的表达式

= sec (3 θ) csc (3 θ)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sec (3 θ) csc (3 θ)

对于

\frac{1}{sec ( 3 θ )} :

将分母和分子乘以

csc (3 θ)

\frac{1}{sec ( 3 θ )} = \frac{1 \cdot csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )} = \frac{csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

对于

\frac{1}{csc ( 3 θ )} :

将分母和分子乘以

sec (3 θ)

\frac{1}{csc ( 3 θ )} = \frac{1 \cdot sec ( 3 θ )}{csc ( 3 θ ) sec ( 3 θ )} = \frac{sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )} - \frac{sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

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{csc ( 3 θ ) - sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{\frac{c s c ( 3 θ ) - s e c ( 3 θ )}{s e c ( 3 θ ) c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ ) c s c ( 3 θ )}}

分式相除:

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

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) sec ( 3 θ ) csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ ) \cdot 1}

整理后得

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) sec ( 3 θ ) csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

约分:

sec (3 θ)

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) csc ( 3 θ )}{csc ( 3 θ )}

约分:

csc (3 θ)

= csc (3 θ) - sec (3 θ)

csc (3 θ) - sec (3 θ)

csc (3 θ) - sec (3 θ)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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