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