Sinus- en cosinusregel

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 \(22^{2} = 20^{2} + 17^{2} - 2 ⋅ 20 ⋅ 17 ⋅ \cos(\angle K)\)
dus \(484 = 689 - 680 ⋅ \cos(\angle K) \text{.}\)

Balansmethode geeft \(\cos(\angle K) = {484 - 689 \over -680} = 0{,}301...\)

Hieruit volgt \(\angle K = \cos^{-1}(0{,}301...) ≈ 72{,}5\degree \text{.}\)

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

Invullen geeft \(27^{2} = 16^{2} + 16^{2} - 2 ⋅ 16 ⋅ 16 ⋅ \cos(\angle Q)\)
dus \(729 = 512 - 512 ⋅ \cos(\angle Q) \text{.}\)

Balansmethode geeft \(\cos(\angle Q) = {729 - 512 \over -512} = -0{,}423...\)

Hieruit volgt \(\angle Q = \cos^{-1}(-0{,}423...) ≈ 115{,}1\degree \text{.}\)

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

Dus \(B\kern{-.8pt}C^{2} = 30^{2} + 35^{2} - 2 ⋅ 30 ⋅ 35 ⋅ \cos(72\degree) = 1476{,}064... \text{.}\)

\(B\kern{-.8pt}C = \sqrt{1476{,}064...} ≈ 38{,}4 \text{.}\)

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

Dus \(K\kern{-.8pt}M^{2} = 38^{2} + 29^{2} - 2 ⋅ 38 ⋅ 29 ⋅ \cos(118\degree) = 3319{,}715... \text{.}\)

\(K\kern{-.8pt}M = \sqrt{3319{,}715...} ≈ 57{,}6 \text{.}\)

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

Daaruit volgt \(\sin(\angle B) = {A\kern{-.8pt}C ⋅ \sin(\angle A) \over B\kern{-.8pt}C} = {25 ⋅ \sin(27\degree) \over 13} = 0{,}873... \text{.}\)

Dit geeft \(\angle B ≈ 60{,}8\degree\) of \(\angle B ≈ 119{,}2\degree \text{.}\)
Uit de afbeelding volgt dat \(\angle B\) een scherpe hoek is, dus \(\angle B ≈ 60{,}8\degree \text{.}\)

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

Daaruit volgt \(\sin(\angle R) = {P\kern{-.8pt}Q ⋅ \sin(\angle Q) \over P\kern{-.8pt}R} = {20 ⋅ \sin(27\degree) \over 10} = 0{,}907... \text{.}\)

Dit geeft \(\angle R ≈ 65{,}2\degree\) of \(\angle R ≈ 114{,}8\degree \text{.}\)
Uit de afbeelding volgt dat \(\angle R\) een stompe hoek is, dus \(\angle R ≈ 114{,}8\degree \text{.}\)

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

Dus \(K\kern{-.8pt}L = {K\kern{-.8pt}M ⋅ \sin(\angle M) \over \sin(\angle L)} = {21 ⋅ \sin(82\degree) \over \sin(46\degree)} \text{.}\)

\(K\kern{-.8pt}L ≈ 28{,}9 \text{.}\)

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

Dus \(Q\kern{-.8pt}R = {P\kern{-.8pt}Q ⋅ \sin(\angle P) \over \sin(\angle R)} = {14 ⋅ \sin(96\degree) \over \sin(43\degree)} \text{.}\)

\(Q\kern{-.8pt}R ≈ 20{,}4 \text{.}\)

Uit \(\angle M + \angle K + \angle L = 180\degree\) volgt \(\angle K = 180\degree - \angle M - \angle L = 180\degree - 65\degree - 27\degree = 88\degree \text{.}\)

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 \(K\kern{-.8pt}L = {L\kern{-.8pt}M ⋅ \sin(\angle M) \over \sin(\angle K)} = {46 ⋅ \sin(65\degree) \over \sin(88\degree)} \text{.}\)

\(K\kern{-.8pt}L ≈ 41{,}7 \text{.}\)

Uit \(\angle L + \angle M + \angle K = 180\degree\) volgt \(\angle M = 180\degree - \angle L - \angle K = 180\degree - 51\degree - 27\degree = 102\degree \text{.}\)

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

Dus \(K\kern{-.8pt}M = {K\kern{-.8pt}L ⋅ \sin(\angle L) \over \sin(\angle M)} = {32 ⋅ \sin(51\degree) \over \sin(102\degree)} \text{.}\)

\(K\kern{-.8pt}M ≈ 25{,}4 \text{.}\)