3. Zadatak

Kvadrirajmo dane jednakosti:

$$\begin{cases} 9\sin^2\alpha+24\sin\alpha\cos\beta+16\cos^2\beta=36,\\ 16\sin^2\beta+24\sin\beta\cos\alpha+9\cos^2\alpha=4. \end{cases}$$

Zbrajanjem ovih jednakosti redom slijedi

$$9+24(\sin\alpha\cos\beta+\sin\beta\cos\alpha)+16=40$$ $$24\sin(\alpha+\beta)=15$$ $$\sin(\alpha+\beta)=\frac58.$$

Kako je treći kut danog trokuta $\gamma=180^\circ-(\alpha+\beta)$ slijedi da je $\sin\gamma=\sin(\alpha+\beta)$.

Dakle, $\sin\gamma=\dfrac58$.

Duljina nasuprotne stranice jest

$$c=2R\sin\gamma=2\cdot6\cdot\frac58=7.5\ \mathrm{cm}.$$