Sinus- en cosinusregel

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

Invullen geeft \(28^2=27^2+26^2-2⋅27⋅26⋅\cos(\angle A)\)
dus \(784=1\,405-1\,404⋅\cos(\angle A)\text{.}\)

Balansmethode geeft \(\cos(\angle A)={784-1\,405 \over -1\,404}=0{,}442...\)

Hieruit volgt \(\angle A=\cos^{-1}(0{,}442...)≈63{,}7\degree\text{.}\)

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 \(40^2=25^2+26^2-2⋅25⋅26⋅\cos(\angle C)\)
dus \(1\,600=1\,301-1\,300⋅\cos(\angle C)\text{.}\)

Balansmethode geeft \(\cos(\angle C)={1\,600-1\,301 \over -1\,300}=-0{,}23\)

Hieruit volgt \(\angle C=\cos^{-1}(-0{,}23)≈103{,}3\degree\text{.}\)

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

Dus \(P\kern{-.8pt}Q^2=34^2+30^2-2⋅34⋅30⋅\cos(68\degree)=1291{,}802...\text{.}\)

\(P\kern{-.8pt}Q=\sqrt{1291{,}802...}≈35{,}9\text{.}\)

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

Dus \(P\kern{-.8pt}Q^2=19^2+16^2-2⋅19⋅16⋅\cos(118\degree)=902{,}438...\text{.}\)

\(P\kern{-.8pt}Q=\sqrt{902{,}438...}≈30{,}0\text{.}\)

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 \(\sin(\angle M)={K\kern{-.8pt}L⋅\sin(\angle L) \over K\kern{-.8pt}M}={23⋅\sin(29\degree) \over 14}=0{,}796...\text{.}\)

Dit geeft \(\angle M≈52{,}8\degree\) of \(\angle M≈127{,}2\degree\text{.}\)
Uit de afbeelding volgt dat \(\angle M\) een scherpe hoek is, dus \(\angle M≈52{,}8\degree\text{.}\)

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 \(\sin(\angle Q)={P\kern{-.8pt}R⋅\sin(\angle P) \over Q\kern{-.8pt}R}={23⋅\sin(25\degree) \over 13}=0{,}747...\text{.}\)

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

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 \(B\kern{-.8pt}C={A\kern{-.8pt}B⋅\sin(\angle A) \over \sin(\angle C)}={16⋅\sin(77\degree) \over \sin(58\degree)}\text{.}\)

\(B\kern{-.8pt}C≈18{,}4\text{.}\)

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

Dus \(A\kern{-.8pt}B={A\kern{-.8pt}C⋅\sin(\angle C) \over \sin(\angle B)}={34⋅\sin(119\degree) \over \sin(35\degree)}\text{.}\)

\(A\kern{-.8pt}B≈51{,}8\text{.}\)

Uit \(\angle L+\angle M+\angle K=180\degree\) volgt \(\angle M=180\degree-\angle L-\angle K=180\degree-62\degree-48\degree=70\degree\text{.}\)

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 \(K\kern{-.8pt}M={K\kern{-.8pt}L⋅\sin(\angle L) \over \sin(\angle M)}={18⋅\sin(62\degree) \over \sin(70\degree)}\text{.}\)

\(K\kern{-.8pt}M≈16{,}9\text{.}\)

Uit \(\angle K+\angle L+\angle M=180\degree\) volgt \(\angle L=180\degree-\angle K-\angle M=180\degree-28\degree-45\degree=107\degree\text{.}\)

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

Dus \(L\kern{-.8pt}M={K\kern{-.8pt}M⋅\sin(\angle K) \over \sin(\angle L)}={61⋅\sin(28\degree) \over \sin(107\degree)}\text{.}\)

\(L\kern{-.8pt}M≈29{,}9\text{.}\)