Aplicatii ale trigonometriei in geometrie

In triunghiul ABC se dau  $\angle A=45\degree,\ b=\sqrt{2},\ c=1$ .
 Atunci latura  $a$   a triunghiului are lungimea:

• 2

• 3

• 1

•  $\sqrt{3-\sqrt{2}}$  

•  $3+\sqrt{6}$  
  

Daca intr-un triunghi ABC raza cercului circumscris este $R=1$ , iar latura $a=1$  , atunci:

• $\angle A=30\degree$

• $\angle A=120\degree$

• $\angle A=45\degree$

• $\angle A\in\left\{60\degree,150\degree\right\}$

• $\angle A\in\left\{30\degree,150\degree\right\}$

Intr-un triunghi ABC avem   $\angle A=90\degree,\ \angle B=15\degree,\ AB=2$  . Atunci latura AC are lungimea :

•  $2tg\ \frac{\pi}{12}$  

•  $2\operatorname{ctg}15\degree$  

•  $2\sin15\degree$  

•  $2\cos\ \frac{\pi}{6}$  

•  $\frac{2}{\cos15\degree}$  

In triunghiul ABC cu   $a=\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{2}$   $B=\frac{\pi}{4},\ C=\frac{\pi}{6}$  , raza R a cercului circumscris este data de:

•  $R=\frac{1}{\sqrt{2}\left(1+\sqrt{3}\right)}$  

•  $R=\frac{1}{1+\sqrt{3}}$  

•  $R=2$  

•  $R=1$  

•  $R=\frac{1+\sqrt{3}}{\sqrt{2}}$  

Daca in triunghiul ABC avem   $AB=\sqrt{13},\ BC=3,\ \angle C=60\degree$  , atunci:

•  $AC=4$  

•  $AC=2$  

• nu exista un asemenea triunghi

•  $AC=4\sqrt{3}$  

•  $AC=\frac{\sqrt{3}}{2}$  

In triunghiul ABC   $\sin C=\cos A+\cos B$  . Atunci:

•  $\Delta ABC$  -echilateral

•  $\angle A=90\degree$  

•  $\angle B=90\degree$  

•  $\angle A=90\degree\ sau\ \angle B=90\degree$  

•  $\angle A=120\degree$  

Fie ABC un triunghi oarecare. Sa se calculeze, in functie de a, b, c, expresia   $E=a\left(b\cos C-c\cos B\right)$  .

•  $E=a\sqrt{b^2+c^2}$  

•  $E=b^2-c^2$  

•  $E=bc$  

•  $E=a\sqrt{bc}$  

•  $E=a^2+b^2+c^2$  

Daca intr-un triunghi avem  $b=3$  , $c=2,\ B+C=\frac{\pi}{3}$  , atunci   $tg\ \frac{B-C}{2}$  este:

•  $5$  

•  $\frac{5\sqrt{3}}{3}$  

•  $\sqrt{3}$  

•  $\frac{\sqrt{3}}{15}$  

•  $\frac{\sqrt{3}}{5}$  

In triunghiul ABC se dau   $AB=3,$   $AC=2,\ \angle A=\frac{2\pi}{3}$  . Sa se determine lungimea medianei AM.

•  $\sqrt{3,5}$  

•  $\sqrt{10}$  

•  $\frac{\sqrt{7}}{2}$  

•  $\frac{\sqrt{19}}{2}$  

•  $\frac{2}{\sqrt{7}}$  

Sa se determine unghiul C al triunghiului ABC  in care are loc relatia:   $\cos A+\cos B=\cos\left(A+B\right)+\frac{3}{2}$  .

•  $10\degree$  

•  $30\degree$  

•  $45\degree$  

•  $60\degree$  

•  $120\degree$