De sinusregel in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \({A\kern{-.8pt}B \over \sin(\angle C)} = {B\kern{-.8pt}C \over \sin(\angle A)} = {A\kern{-.8pt}C \over \sin(\angle B)} \text{.}\)

Dus \(B\kern{-.8pt}C = {A\kern{-.8pt}B ⋅ \sin(\angle A) \over \sin(\angle C)} = {36 ⋅ \sin(88\degree) \over \sin(58\degree)} \text{.}\)

\(B\kern{-.8pt}C ≈ 42{,}4 \text{.}\)

De sinusregel in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \({Q\kern{-.8pt}R \over \sin(\angle P)} = {P\kern{-.8pt}R \over \sin(\angle Q)} = {P\kern{-.8pt}Q \over \sin(\angle R)} \text{.}\)

Dus \(P\kern{-.8pt}R = {Q\kern{-.8pt}R ⋅ \sin(\angle Q) \over \sin(\angle P)} = {26 ⋅ \sin(105\degree) \over \sin(25\degree)} \text{.}\)

\(P\kern{-.8pt}R ≈ 59{,}4 \text{.}\)

De sinusregel in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \({A\kern{-.8pt}B \over \sin(\angle C)} = {B\kern{-.8pt}C \over \sin(\angle A)} = {A\kern{-.8pt}C \over \sin(\angle B)} \text{.}\)

Daaruit volgt \(\sin(\angle A) = {B\kern{-.8pt}C ⋅ \sin(\angle C) \over A\kern{-.8pt}B} = {17 ⋅ \sin(34\degree) \over 12} = 0{,}792... \text{.}\)

Dit geeft \(\angle A ≈ 52{,}4\degree\) of \(\angle A ≈ 127{,}6\degree \text{.}\)
Uit de afbeelding volgt dat \(\angle A\) een scherpe hoek is, dus \(\angle A ≈ 52{,}4\degree \text{.}\)

De sinusregel in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \({K\kern{-.8pt}M \over \sin(\angle L)} = {K\kern{-.8pt}L \over \sin(\angle M)} = {L\kern{-.8pt}M \over \sin(\angle K)} \text{.}\)

Daaruit volgt \(\sin(\angle M) = {K\kern{-.8pt}L ⋅ \sin(\angle L) \over K\kern{-.8pt}M} = {15 ⋅ \sin(37\degree) \over 11} = 0{,}820... \text{.}\)

Dit geeft \(\angle M ≈ 55{,}2\degree\) of \(\angle M ≈ 124{,}8\degree \text{.}\)
Uit de afbeelding volgt dat \(\angle M\) een stompe hoek is, dus \(\angle M ≈ 124{,}8\degree \text{.}\)

Uit \(\angle M + \angle K + \angle L = 180\degree\) volgt \(\angle K = 180\degree - \angle M - \angle L = 180\degree - 61\degree - 48\degree = 71\degree \text{.}\)

De sinusregel in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \({K\kern{-.8pt}L \over \sin(\angle M)} = {L\kern{-.8pt}M \over \sin(\angle K)} = {K\kern{-.8pt}M \over \sin(\angle L)} \text{.}\)

Dus \(K\kern{-.8pt}L = {L\kern{-.8pt}M ⋅ \sin(\angle M) \over \sin(\angle K)} = {38 ⋅ \sin(61\degree) \over \sin(71\degree)} \text{.}\)

\(K\kern{-.8pt}L ≈ 35{,}2 \text{.}\)

Uit \(\angle P + \angle Q + \angle R = 180\degree\) volgt \(\angle Q = 180\degree - \angle P - \angle R = 180\degree - 29\degree - 34\degree = 117\degree \text{.}\)

De sinusregel in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \({Q\kern{-.8pt}R \over \sin(\angle P)} = {P\kern{-.8pt}R \over \sin(\angle Q)} = {P\kern{-.8pt}Q \over \sin(\angle R)} \text{.}\)

Dus \(Q\kern{-.8pt}R = {P\kern{-.8pt}R ⋅ \sin(\angle P) \over \sin(\angle Q)} = {23 ⋅ \sin(29\degree) \over \sin(117\degree)} \text{.}\)

\(Q\kern{-.8pt}R ≈ 12{,}5 \text{.}\)

De cosinusregel in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(A\kern{-.8pt}C^{2} = A\kern{-.8pt}B^{2} + B\kern{-.8pt}C^{2} - 2 ⋅ A\kern{-.8pt}B ⋅ B\kern{-.8pt}C ⋅ \cos(\angle B) \text{.}\)

Dus \(A\kern{-.8pt}C^{2} = 25^{2} + 22^{2} - 2 ⋅ 25 ⋅ 22 ⋅ \cos(81\degree) = 936{,}922... \text{.}\)

\(A\kern{-.8pt}C = \sqrt{936{,}922...} ≈ 30{,}6 \text{.}\)

De cosinusregel in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(L\kern{-.8pt}M^{2} = K\kern{-.8pt}M^{2} + K\kern{-.8pt}L^{2} - 2 ⋅ K\kern{-.8pt}M ⋅ K\kern{-.8pt}L ⋅ \cos(\angle K) \text{.}\)

Dus \(L\kern{-.8pt}M^{2} = 30^{2} + 26^{2} - 2 ⋅ 30 ⋅ 26 ⋅ \cos(95\degree) = 1711{,}962... \text{.}\)

\(L\kern{-.8pt}M = \sqrt{1711{,}962...} ≈ 41{,}4 \text{.}\)

De cosinusregel in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(L\kern{-.8pt}M^{2} = K\kern{-.8pt}M^{2} + K\kern{-.8pt}L^{2} - 2 ⋅ K\kern{-.8pt}M ⋅ K\kern{-.8pt}L ⋅ \cos(\angle K) \text{.}\)

Invullen geeft \(54^{2} = 48^{2} + 28^{2} - 2 ⋅ 48 ⋅ 28 ⋅ \cos(\angle K)\)
dus \(2\,916 = 3\,088 - 2\,688 ⋅ \cos(\angle K) \text{.}\)

Balansmethode geeft \(\cos(\angle K) = {2\,916 - 3\,088 \over -2\,688} = 0{,}063...\)

Hieruit volgt \(\angle K = \cos^{-1}(0{,}063...) ≈ 86{,}3\degree \text{.}\)

De cosinusregel in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(P\kern{-.8pt}Q^{2} = Q\kern{-.8pt}R^{2} + P\kern{-.8pt}R^{2} - 2 ⋅ Q\kern{-.8pt}R ⋅ P\kern{-.8pt}R ⋅ \cos(\angle R) \text{.}\)

Invullen geeft \(28^{2} = 20^{2} + 18^{2} - 2 ⋅ 20 ⋅ 18 ⋅ \cos(\angle R)\)
dus \(784 = 724 - 720 ⋅ \cos(\angle R) \text{.}\)

Balansmethode geeft \(\cos(\angle R) = {784 - 724 \over -720} = -0{,}083...\)

Hieruit volgt \(\angle R = \cos^{-1}(-0{,}083...) ≈ 94{,}8\degree \text{.}\)