Question

a) Sendo Tgx= 1/2 determina o valor de Tg2x

b) Sen2x, sendo Senx - Cosx = 1/5

Answer 1

Boa noite Narica

tg(2x) = 2*tg(x)/(1 - tg²(x))
tg(2x) = 2*(1/2)/(1 - 1/4) 
tg(2x) = 1/(3/4) = 4/3 

sen(x) - cos(x) = 1/5 

(sen(x) - cos(x))² = (1/5)²

sen²(x) - 2sen(x)*cos(x) + cos²(x) = 1/25

1 - sen(2x) = 1/25

sen(2x) = 25/25 - 1/25 = 24/25

Answer 2

a)

$\tan(2x) = \frac{2 \tan(x) }{1 - { \tan}^{2}x } = \frac{2. \frac{1}{2} }{1 - { (\frac{1}{2}) }^{2} }$

$\tan(2x) = \frac{1}{1 - \frac{1}{4} } = \frac{1}{ \frac{3}{4} } = \frac{4}{3}$

b)

$\sin(x)- \cos(x) = \frac{1}{5} \\ {( \sin(x) - \cos(x) )}^{2} = {( \frac{1}{5} )}^{2}$

${ \sin}^{2}x - 2 \sin(x) \cos(x) \\ + { \cos }^{2} x = \frac{1}{25} \\ 1 - \sin(2x) = \frac{1}{25}$

$\sin(2x) = 1 - \frac{1}{25} \\ \sin(2x) = \frac{25 - 1}{25} = \frac{24}{25}$