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受欢迎的三角函数 >

证明 (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ)

解答

证明

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

解答

真

求解步骤

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

调整左侧

\frac{sin ( 2 θ )}{sin ( θ )} - \frac{cos ( 2 θ )}{cos ( θ )}

化简

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

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

cos (θ), sin (θ)

的最小公倍数:

cos (θ) sin (θ)

cos (θ), sin (θ)

最小公倍数 (LCM)

计算出由出现在

cos (θ)

或

sin (θ)

中的因子组成的表达式

= cos (θ) sin (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

cos (θ) sin (θ)

对于

\frac{cos ( 2 θ )}{cos ( θ )} :

将分母和分子乘以

sin (θ)

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

对于

\frac{sin ( 2 θ )}{sin ( θ )} :

将分母和分子乘以

cos (θ)

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

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

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

使用三角恒等式改写

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

使用角差恒等式:

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

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

化简

\frac{sin ( 2 θ - θ )}{cos ( θ ) sin ( θ )} : \frac{1}{cos ( θ )}

\frac{sin ( 2 θ - θ )}{cos ( θ ) sin ( θ )}

同类项相加:

2 θ - θ = θ

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

约分:

sin (θ)

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{\frac{1}{s e c ( θ )}}

化简

\frac{1}{\frac{1}{s e c ( θ )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ( θ )}{1}

使用法则

\frac{a}{1} = a

= sec (θ)

sec (θ)

sec (θ)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 sec(2θ)=(sec^2(θ))/(1-tan^2(θ))

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

证明 cos(2a)=cos^2(a)-sin^2(a)

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

证明 tan(θ)+cot(θ)=sec(θ)*csc(θ)

p ro v e tan (θ) + cot (θ) = sec (θ) \cdot csc (θ)

证明 (cos(x))/(csc(x)+1)+(cos(x))/(csc(x)-1)=2tan(x)

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

证明 (cos(x)csc(x))/(cot^2(x))=tan(x)

p ro v e \frac{cos ( x ) csc ( x )}{cot ^{2} ( x )} = tan (x)