Question

Вычислите:

$1-cos(2a)$

если $sin(a)=frac{1}{sqrt{5}}$

Answer 1

sinα=1/√5
I способ.
$boxed{cos2 alpha = 1-2sin^2 alpha}$
$displaystyle 1-cos2 alpha=1-(1-2sin^2 alpha)=1-(1-2 cdot (frac{1}{ sqrt 5})^2)=1-(1- frac{2}{5})=$
$displaystyle = 1- frac{3}{5}= frac{2}{5}$
Ответ: 2/5

II способ.
$boxed{cos 2 alpha = cos^2 alpha - sin^2 alpha}$
$displaystyle sin alpha = frac{1}{ sqrt 5} Rightarrow cos alpha = sqrt{1- (frac{1}{ sqrt 5})^2}= sqrt{1- frac{1}{5}}= sqrt{ frac{4}{5}}= frac{2}{ sqrt 5}$
$displaystyle 1-cos2 alpha = 1-( (frac{2}{ sqrt 5})^2-(frac{1}{ sqrt 5})^2)=1-( frac{4}{5}- frac{1}{5})=1- frac{3}{5}= frac{2}{5}$
Ответ: 2/5

III способ.
$boxed{cos2 alpha = 2cos^2 alpha - 1}$
$displaystyle sin alpha = frac{1}{ sqrt 5} Rightarrow cos alpha = frac{2}{ sqrt 5}$
$displaystyle 1-cos2 alpha = 1-(2cos^2 alpha -1)=1-( 2 cdot (frac{2}{ sqrt 5})^2-1)=$
$displaystyle = 1- (2 cdot frac{4}{5}-1)=1- frac{8-5}{5}=1- frac{3}{5}= frac{2}{5}$
Ответ: 2/5