Question Number 193522 by Mingma last updated on 16/Jun/23
Answered by Subhi last updated on 16/Jun/23
$$\frac{{x}+{z}}{{sin}\left(\mathrm{100}\right)}=\frac{{x}}{{sin}\left(\mathrm{40}\right)} \\ $$$${sin}\left(\mathrm{100}\right){x}={sin}\left(\mathrm{40}\right){x}+{sin}\left(\mathrm{40}\right){z}\:\Rrightarrow\:{x}=\frac{{sin}\left(\mathrm{40}\right){z}}{{sin}\left(\mathrm{10}\right)−{sin}\left(\mathrm{40}\right)}\:\left({i}\right) \\ $$$$\frac{{z}}{{sin}\left({y}−\mathrm{40}\right)}=\frac{{x}+{z}}{{sin}\left({u}\right)} \\ $$$${u}+{y}−\mathrm{40}=\mathrm{180}−\mathrm{140}=\mathrm{40}\:\Rrightarrow\:{y}=\mathrm{80}−{u}\:\left({ii}\right) \\ $$$$\frac{{z}}{{sin}\left(\mathrm{40}−{u}\right)}=\frac{{x}+{z}}{{sin}\left({u}\right)} \\ $$$${sin}\left(\mathrm{40}−{u}\right){x}+{sin}\left(\mathrm{40}−{u}\right){z}={zsin}\left({u}\right) \\ $$$${sin}\left(\mathrm{40}−{u}\right).\frac{{sin}\left(\mathrm{40}\right){z}}{{sin}\left(\mathrm{100}\right)−{sin}\left(\mathrm{40}\right)}+{sin}\left(\mathrm{40}−{u}\right){z}={zsin}\left({u}\right) \\ $$$$\left(\frac{{sin}^{\mathrm{2}} \left(\mathrm{40}\right)}{{sin}\left(\mathrm{100}\right)−{sin}\left(\mathrm{40}\right)}+{sin}\left(\mathrm{40}\right)\right){cos}\left({u}\right)=\left(\mathrm{1}+{cos}\left(\mathrm{40}\right)+\frac{{cos}\left(\mathrm{40}\right).{sin}\left(\mathrm{40}\right)}{{sin}\left(\mathrm{100}\right)−{sin}\left(\mathrm{40}\right)}\right){sin}\left({u}\right) \\ $$$${u}={tan}^{−\mathrm{1}} \left(\left(\frac{{sin}^{\mathrm{2}} \left(\mathrm{40}\right)}{{sin}\left(\mathrm{100}\right)−{sin}\left(\mathrm{40}\right)}+{sin}\left(\mathrm{40}\right)\right)\right)/\left(\mathrm{1}+{cos}\left(\mathrm{40}\right)+\frac{\frac{{sin}\left(\mathrm{80}\right)}{\mathrm{2}}}{{sin}\left(\mathrm{100}\right)−{sin}\left(\mathrm{40}\right)}\right)={tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)=\mathrm{30} \\ $$
Commented by Subhi last updated on 16/Jun/23
Commented by Mingma last updated on 16/Jun/23
Perfect