Getal & Ruimte (13e editie) - havo wiskunde B
'Sinus, cosinus en tangens'.
| 3 havo | 6.3 Berekeningen met de tangens |
opgave 13p a Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=38\text{,}\) \(\angle B=41\degree\) en \(\angle C=90\degree\text{.}\) Tangens (1) 007m - Sinus, cosinus en tangens - basis - 0ms a Tangens in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\tan(\angle B)={A\kern{-.8pt}C \over B\kern{-.8pt}C}\) ofwel \(\tan(41\degree)={A\kern{-.8pt}C \over 38}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}C=38⋅\tan(41\degree)\text{.}\) 1p ○ Dus \(A\kern{-.8pt}C≈33{,}0\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}C=22\text{,}\) \(\angle B=31\degree\) en \(\angle C=90\degree\text{.}\) Tangens (2) 007n - Sinus, cosinus en tangens - basis - 0ms b Tangens in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\tan(\angle B)={A\kern{-.8pt}C \over B\kern{-.8pt}C}\) ofwel \(\tan(31\degree)={22 \over B\kern{-.8pt}C}\text{.}\) 1p ○ Hieruit volgt \(B\kern{-.8pt}C={22 \over \tan(31\degree)}\text{.}\) 1p ○ Dus \(B\kern{-.8pt}C≈36{,}6\text{.}\) 1p 3p c Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}L=47\text{,}\) \(L\kern{-.8pt}M=25\) en \(\angle L=90\degree\text{.}\) Tangens (3) 007o - Sinus, cosinus en tangens - basis - 0ms c Tangens in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\tan(\angle K)={L\kern{-.8pt}M \over K\kern{-.8pt}L}\) ofwel \(\tan(\angle K)={25 \over 47}\text{.}\) 1p ○ Hieruit volgt \(\angle K=\tan^{-1}({25 \over 47})\text{.}\) 1p ○ Dus \(\angle K≈28{,}0\degree\text{.}\) 1p |
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| 3 havo | 6.4 De sinus en de cosinus |
opgave 13p a Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=75\text{,}\) \(\angle P=41\degree\) en \(\angle Q=90\degree\text{.}\) Sinus (1) 007g - Sinus, cosinus en tangens - basis - 0ms a Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle P)={Q\kern{-.8pt}R \over P\kern{-.8pt}R}\) ofwel \(\sin(41\degree)={Q\kern{-.8pt}R \over 75}\text{.}\) 1p ○ Hieruit volgt \(Q\kern{-.8pt}R=75⋅\sin(41\degree)\text{.}\) 1p ○ Dus \(Q\kern{-.8pt}R≈49{,}2\text{.}\) 1p 3p b Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(Q\kern{-.8pt}R=60\text{,}\) \(\angle P=42\degree\) en \(\angle Q=90\degree\text{.}\) Sinus (2) 007h - Sinus, cosinus en tangens - basis - 0ms b Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle P)={Q\kern{-.8pt}R \over P\kern{-.8pt}R}\) ofwel \(\sin(42\degree)={60 \over P\kern{-.8pt}R}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R={60 \over \sin(42\degree)}\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈89{,}7\text{.}\) 1p 3p c Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(L\kern{-.8pt}M=60\text{,}\) \(K\kern{-.8pt}M=67\) en \(\angle L=90\degree\text{.}\) Sinus (3) 007i - Sinus, cosinus en tangens - basis - 0ms c Sinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\sin(\angle K)={L\kern{-.8pt}M \over K\kern{-.8pt}M}\) ofwel \(\sin(\angle K)={60 \over 67}\text{.}\) 1p ○ Hieruit volgt \(\angle K=\sin^{-1}({60 \over 67})\text{.}\) 1p ○ Dus \(\angle K≈63{,}6\degree\text{.}\) 1p 3p d Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}L=49\text{,}\) \(\angle L=32\degree\) en \(\angle M=90\degree\text{.}\) Cosinus (1) 007j - Sinus, cosinus en tangens - basis - 0ms d Cosinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\cos(\angle L)={L\kern{-.8pt}M \over K\kern{-.8pt}L}\) ofwel \(\cos(32\degree)={L\kern{-.8pt}M \over 49}\text{.}\) 1p ○ Hieruit volgt \(L\kern{-.8pt}M=49⋅\cos(32\degree)\text{.}\) 1p ○ Dus \(L\kern{-.8pt}M≈41{,}6\text{.}\) 1p opgave 23p a Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(Q\kern{-.8pt}R=24\text{,}\) \(\angle Q=47\degree\) en \(\angle R=90\degree\text{.}\) Cosinus (2) 007k - Sinus, cosinus en tangens - basis - 0ms a Cosinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\cos(\angle Q)={Q\kern{-.8pt}R \over P\kern{-.8pt}Q}\) ofwel \(\cos(47\degree)={24 \over P\kern{-.8pt}Q}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}Q={24 \over \cos(47\degree)}\text{.}\) 1p ○ Dus \(P\kern{-.8pt}Q≈35{,}2\text{.}\) 1p 3p b Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}L=42\text{,}\) \(K\kern{-.8pt}M=47\) en \(\angle L=90\degree\text{.}\) Cosinus (3) 007l - Sinus, cosinus en tangens - basis - 0ms b Cosinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\cos(\angle K)={K\kern{-.8pt}L \over K\kern{-.8pt}M}\) ofwel \(\cos(\angle K)={42 \over 47}\text{.}\) 1p ○ Hieruit volgt \(\angle K=\cos^{-1}({42 \over 47})\text{.}\) 1p ○ Dus \(\angle K≈26{,}7\degree\text{.}\) 1p |