Sinus- en cosinusregel

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

Invullen geeft \(27^{2} = 14^{2} + 24^{2} - 2 ⋅ 14 ⋅ 24 ⋅ \cos(\angle C)\)
dus \(729 = 772 - 672 ⋅ \cos(\angle C) \text{.}\)

Balansmethode geeft \(\cos(\angle C) = {729 - 772 \over -672} = 0{,}063...\)

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

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

Invullen geeft \(36^{2} = 24^{2} + 20^{2} - 2 ⋅ 24 ⋅ 20 ⋅ \cos(\angle P)\)
dus \(1\,296 = 976 - 960 ⋅ \cos(\angle P) \text{.}\)

Balansmethode geeft \(\cos(\angle P) = {1\,296 - 976 \over -960} = -0{,}333...\)

Hieruit volgt \(\angle P = \cos^{-1}(-0{,}333...) ≈ 109{,}5\degree \text{.}\)

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

Dus \(K\kern{-.8pt}L^{2} = 17^{2} + 19^{2} - 2 ⋅ 17 ⋅ 19 ⋅ \cos(75\degree) = 482{,}802... \text{.}\)

\(K\kern{-.8pt}L = \sqrt{482{,}802...} ≈ 22{,}0 \text{.}\)

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 \(K\kern{-.8pt}M^{2} = 27^{2} + 33^{2} - 2 ⋅ 27 ⋅ 33 ⋅ \cos(99\degree) = 2096{,}766... \text{.}\)

\(K\kern{-.8pt}M = \sqrt{2096{,}766...} ≈ 45{,}8 \text{.}\)

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

Daaruit volgt \(\sin(\angle P) = {Q\kern{-.8pt}R ⋅ \sin(\angle R) \over P\kern{-.8pt}Q} = {29 ⋅ \sin(26\degree) \over 17} = 0{,}747... \text{.}\)

Dit geeft \(\angle P ≈ 48{,}4\degree\) of \(\angle P ≈ 131{,}6\degree \text{.}\)
Uit de afbeelding volgt dat \(\angle P\) een scherpe hoek is, dus \(\angle P ≈ 48{,}4\degree \text{.}\)

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

Daaruit volgt \(\sin(\angle Q) = {P\kern{-.8pt}R ⋅ \sin(\angle P) \over Q\kern{-.8pt}R} = {24 ⋅ \sin(39\degree) \over 17} = 0{,}888... \text{.}\)

Dit geeft \(\angle Q ≈ 62{,}7\degree\) of \(\angle Q ≈ 117{,}3\degree \text{.}\)
Uit de afbeelding volgt dat \(\angle Q\) een stompe hoek is, dus \(\angle Q ≈ 117{,}3\degree \text{.}\)

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 \(K\kern{-.8pt}L = {K\kern{-.8pt}M ⋅ \sin(\angle M) \over \sin(\angle L)} = {12 ⋅ \sin(54\degree) \over \sin(65\degree)} \text{.}\)

\(K\kern{-.8pt}L ≈ 10{,}7 \text{.}\)

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

Dus \(P\kern{-.8pt}Q = {P\kern{-.8pt}R ⋅ \sin(\angle R) \over \sin(\angle Q)} = {20 ⋅ \sin(105\degree) \over \sin(35\degree)} \text{.}\)

\(P\kern{-.8pt}Q ≈ 33{,}7 \text{.}\)

Uit \(\angle C + \angle A + \angle B = 180\degree\) volgt \(\angle A = 180\degree - \angle C - \angle B = 180\degree - 49\degree - 44\degree = 87\degree \text{.}\)

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

Dus \(A\kern{-.8pt}B = {B\kern{-.8pt}C ⋅ \sin(\angle C) \over \sin(\angle A)} = {37 ⋅ \sin(49\degree) \over \sin(87\degree)} \text{.}\)

\(A\kern{-.8pt}B ≈ 28{,}0 \text{.}\)

Uit \(\angle P + \angle Q + \angle R = 180\degree\) volgt \(\angle Q = 180\degree - \angle P - \angle R = 180\degree - 25\degree - 37\degree = 118\degree \text{.}\)

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 \(Q\kern{-.8pt}R = {P\kern{-.8pt}R ⋅ \sin(\angle P) \over \sin(\angle Q)} = {31 ⋅ \sin(25\degree) \over \sin(118\degree)} \text{.}\)

\(Q\kern{-.8pt}R ≈ 14{,}8 \text{.}\)