Question

Buna! Am nevoie de rezolvarea acestui exercitiu. Dau 70 de puncte!

Mi-ar fi mai usor sa inteleg rezolvarea dintr-o poza, dar nu conteaza asa mult.

Answer 1

Explicație pas cu pas:

$a, b \in \Big(0 ; \frac{\pi}{2} \Big), \ \ cadranul \ 1$

a)

$\boxed {\sin^{2} a + \cos^{2} a = 1}$

$\cos^{2} a = 1 - \sin^{2} a = 1 - {\Big(\frac{4}{5} \Big)}^{2} = 1 - \frac{16}{25} = \frac{9}{25} \\ \cos a = \pm \frac{3}{5} \implies \cos a = \frac{3}{5}$

b)

$\boxed {\sin (a - b) = \sin a \cdot \cos b - \cos a \cdot \sin b}$

$\sin^{2} b = 1 - \cos^{2} b = 1 - {\Big(\frac{3}{5} \Big)}^{2} = 1 - \frac{9}{25} = \frac{16}{25} \\ \sin b = \pm \frac{4}{5} \implies \sin b = \frac{4}{5}$

$\sin (a - b) = \frac{4}{5} \cdot \frac{3}{5} - \frac{3}{5} \cdot \frac{4}{5} = \frac{12}{25} - \frac{12}{25} = 0$

c)

$\sin 2a + \cos 2a = 2 \sin a \cos a + 1 - 2 \sin^{2} a = \\ = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} + 1 - 2 \cdot \frac{16}{25} = \frac{24 + 25 - 32}{25} = \frac{17}{25} < 2$