Getal & Ruimte (13e editie) - havo wiskunde B
'Sinus, cosinus en tangens'.
| 3 havo | 6.3 Berekeningen met de tangens |
opgave 13p a Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(Q\kern{-.8pt}R=51\text{,}\) \(\angle Q=54\degree\) en \(\angle R=90\degree\text{.}\) Tangens (1) 007m - Sinus, cosinus en tangens - basis - 0ms a Tangens in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\tan(\angle Q)={P\kern{-.8pt}R \over Q\kern{-.8pt}R}\) ofwel \(\tan(54\degree)={P\kern{-.8pt}R \over 51}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R=51⋅\tan(54\degree)\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈70{,}2\text{.}\) 1p 3p b Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=20\text{,}\) \(\angle Q=37\degree\) en \(\angle R=90\degree\text{.}\) Tangens (2) 007n - Sinus, cosinus en tangens - basis - 0ms b Tangens in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\tan(\angle Q)={P\kern{-.8pt}R \over Q\kern{-.8pt}R}\) ofwel \(\tan(37\degree)={20 \over Q\kern{-.8pt}R}\text{.}\) 1p ○ Hieruit volgt \(Q\kern{-.8pt}R={20 \over \tan(37\degree)}\text{.}\) 1p ○ Dus \(Q\kern{-.8pt}R≈26{,}5\text{.}\) 1p 3p c Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}L=34\text{,}\) \(L\kern{-.8pt}M=50\) 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 \(\tan(\angle K)={50 \over 34}\text{.}\) 1p ○ Hieruit volgt \(\angle K=\tan^{-1}({50 \over 34})\text{.}\) 1p ○ Dus \(\angle K≈55{,}8\degree\text{.}\) 1p |
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| 3 havo | 6.4 De sinus en de cosinus |
opgave 13p a Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(L\kern{-.8pt}M=79\text{,}\) \(\angle M=59\degree\) en \(\angle K=90\degree\text{.}\) Sinus (1) 007g - Sinus, cosinus en tangens - basis - 0ms a Sinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\sin(\angle M)={K\kern{-.8pt}L \over L\kern{-.8pt}M}\) ofwel \(\sin(59\degree)={K\kern{-.8pt}L \over 79}\text{.}\) 1p ○ Hieruit volgt \(K\kern{-.8pt}L=79⋅\sin(59\degree)\text{.}\) 1p ○ Dus \(K\kern{-.8pt}L≈67{,}7\text{.}\) 1p 3p b Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=30\text{,}\) \(\angle Q=50\degree\) en \(\angle R=90\degree\text{.}\) Sinus (2) 007h - Sinus, cosinus en tangens - basis - 0ms b Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle Q)={P\kern{-.8pt}R \over P\kern{-.8pt}Q}\) ofwel \(\sin(50\degree)={30 \over P\kern{-.8pt}Q}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}Q={30 \over \sin(50\degree)}\text{.}\) 1p ○ Dus \(P\kern{-.8pt}Q≈39{,}2\text{.}\) 1p 3p c Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=55\text{,}\) \(P\kern{-.8pt}Q=69\) en \(\angle R=90\degree\text{.}\) Sinus (3) 007i - Sinus, cosinus en tangens - basis - 0ms c Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle Q)={P\kern{-.8pt}R \over P\kern{-.8pt}Q}\) ofwel \(\sin(\angle Q)={55 \over 69}\text{.}\) 1p ○ Hieruit volgt \(\angle Q=\sin^{-1}({55 \over 69})\text{.}\) 1p ○ Dus \(\angle Q≈52{,}9\degree\text{.}\) 1p 3p d Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=65\text{,}\) \(\angle P=41\degree\) en \(\angle Q=90\degree\text{.}\) Cosinus (1) 007j - Sinus, cosinus en tangens - basis - 0ms d Cosinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\cos(\angle P)={P\kern{-.8pt}Q \over P\kern{-.8pt}R}\) ofwel \(\cos(41\degree)={P\kern{-.8pt}Q \over 65}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}Q=65⋅\cos(41\degree)\text{.}\) 1p ○ Dus \(P\kern{-.8pt}Q≈49{,}1\text{.}\) 1p opgave 23p a Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}Q=30\text{,}\) \(\angle P=35\degree\) en \(\angle Q=90\degree\text{.}\) Cosinus (2) 007k - Sinus, cosinus en tangens - basis - 0ms a Cosinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\cos(\angle P)={P\kern{-.8pt}Q \over P\kern{-.8pt}R}\) ofwel \(\cos(35\degree)={30 \over P\kern{-.8pt}R}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R={30 \over \cos(35\degree)}\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈36{,}6\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}C=26\text{,}\) \(B\kern{-.8pt}C=44\) en \(\angle A=90\degree\text{.}\) Cosinus (3) 007l - Sinus, cosinus en tangens - basis - 0ms b Cosinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\cos(\angle C)={A\kern{-.8pt}C \over B\kern{-.8pt}C}\) ofwel \(\cos(\angle C)={26 \over 44}\text{.}\) 1p ○ Hieruit volgt \(\angle C=\cos^{-1}({26 \over 44})\text{.}\) 1p ○ Dus \(\angle C≈53{,}8\degree\text{.}\) 1p |