Sinus- en cosinusregel

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 \(36^2=20^2+33^2-2⋅20⋅33⋅\cos(\angle Q)\)
dus \(1\,296=1\,489-1\,320⋅\cos(\angle Q)\text{.}\)

Balansmethode geeft \(\cos(\angle Q)={1\,296-1\,489 \over -1\,320}=0{,}146...\)

Hieruit volgt \(\angle Q=\cos^{-1}(0{,}146...)≈81{,}6\degree\text{.}\)

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

Invullen geeft \(25^2=14^2+21^2-2⋅14⋅21⋅\cos(\angle B)\)
dus \(625=637-588⋅\cos(\angle B)\text{.}\)

Balansmethode geeft \(\cos(\angle B)={625-637 \over -588}=0{,}020...\)

Hieruit volgt \(\angle B=\cos^{-1}(0{,}020...)≈88{,}8\degree\text{.}\)

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 \(K\kern{-.8pt}M^2=22^2+18^2-2⋅22⋅18⋅\cos(77\degree)=629{,}838...\text{.}\)

\(K\kern{-.8pt}M=\sqrt{629{,}838...}≈25{,}1\text{.}\)

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 \(A\kern{-.8pt}C^2=17^2+15^2-2⋅17⋅15⋅\cos(118\degree)=753{,}430...\text{.}\)

\(A\kern{-.8pt}C=\sqrt{753{,}430...}≈27{,}4\text{.}\)

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

Daaruit volgt \(\sin(\angle C)={A\kern{-.8pt}B⋅\sin(\angle B) \over A\kern{-.8pt}C}={29⋅\sin(28\degree) \over 17}=0{,}800...\text{.}\)

Dit geeft \(\angle C≈53{,}2\degree\) of \(\angle C≈126{,}8\degree\text{.}\)
Uit de afbeelding volgt dat \(\angle C\) een scherpe hoek is, dus \(\angle C≈53{,}2\degree\text{.}\)

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

Daaruit volgt \(\sin(\angle A)={B\kern{-.8pt}C⋅\sin(\angle C) \over A\kern{-.8pt}B}={26⋅\sin(26\degree) \over 13}=0{,}876...\text{.}\)

Dit geeft \(\angle A≈61{,}3\degree\) of \(\angle A≈118{,}7\degree\text{.}\)
Uit de afbeelding volgt dat \(\angle A\) een stompe hoek is, dus \(\angle A≈118{,}7\degree\text{.}\)

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 \(L\kern{-.8pt}M={K\kern{-.8pt}L⋅\sin(\angle K) \over \sin(\angle M)}={20⋅\sin(82\degree) \over \sin(52\degree)}\text{.}\)

\(L\kern{-.8pt}M≈25{,}1\text{.}\)

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 \(Q\kern{-.8pt}R={P\kern{-.8pt}Q⋅\sin(\angle P) \over \sin(\angle R)}={21⋅\sin(113\degree) \over \sin(34\degree)}\text{.}\)

\(Q\kern{-.8pt}R≈34{,}6\text{.}\)

Uit \(\angle Q+\angle R+\angle P=180\degree\) volgt \(\angle R=180\degree-\angle Q-\angle P=180\degree-61\degree-55\degree=64\degree\text{.}\)

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

Dus \(P\kern{-.8pt}R={P\kern{-.8pt}Q⋅\sin(\angle Q) \over \sin(\angle R)}={32⋅\sin(61\degree) \over \sin(64\degree)}\text{.}\)

\(P\kern{-.8pt}R≈31{,}1\text{.}\)

Uit \(\angle R+\angle P+\angle Q=180\degree\) volgt \(\angle P=180\degree-\angle R-\angle Q=180\degree-29\degree-32\degree=119\degree\text{.}\)

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 \(P\kern{-.8pt}Q={Q\kern{-.8pt}R⋅\sin(\angle R) \over \sin(\angle P)}={43⋅\sin(29\degree) \over \sin(119\degree)}\text{.}\)

\(P\kern{-.8pt}Q≈23{,}8\text{.}\)