Getal & Ruimte (12e editie) - vwo wiskunde B
'Sinus, cosinus en tangens'.
| 3 vwo | 6.3 Berekeningen met de tangens |
opgave 13p a Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=24\text{,}\) \(\angle B=54\degree\) 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 \(\tan(54\degree)={A\kern{-.8pt}C \over 24}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}C=24⋅\tan(54\degree)\text{.}\) 1p ○ Dus \(A\kern{-.8pt}C≈33{,}0\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=30\text{,}\) \(\angle A=58\degree\) en \(\angle B=90\degree\text{.}\) Tangens (2) 007n - Sinus, cosinus en tangens - basis - 0ms b Tangens in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\tan(\angle A)={B\kern{-.8pt}C \over A\kern{-.8pt}B}\) ofwel \(\tan(58\degree)={30 \over A\kern{-.8pt}B}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}B={30 \over \tan(58\degree)}\text{.}\) 1p ○ Dus \(A\kern{-.8pt}B≈18{,}7\text{.}\) 1p 3p c Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=30\text{,}\) \(P\kern{-.8pt}Q=34\) en \(\angle P=90\degree\text{.}\) Tangens (3) 007o - Sinus, cosinus en tangens - basis - 0ms c Tangens in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\tan(\angle R)={P\kern{-.8pt}Q \over P\kern{-.8pt}R}\) ofwel \(\tan(\angle R)={34 \over 30}\text{.}\) 1p ○ Hieruit volgt \(\angle R=\tan^{-1}({34 \over 30})\text{.}\) 1p ○ Dus \(\angle R≈48{,}6\degree\text{.}\) 1p |
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| 3 vwo | 6.4 De sinus en de cosinus |
opgave 13p a Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(L\kern{-.8pt}M=58\text{,}\) \(\angle M=36\degree\) 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 \(\sin(36\degree)={K\kern{-.8pt}L \over 58}\text{.}\) 1p ○ Hieruit volgt \(K\kern{-.8pt}L=58⋅\sin(36\degree)\text{.}\) 1p ○ Dus \(K\kern{-.8pt}L≈34{,}1\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}B=51\text{,}\) \(\angle C=31\degree\) en \(\angle A=90\degree\text{.}\) Sinus (2) 007h - Sinus, cosinus en tangens - basis - 0ms b Sinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\sin(\angle C)={A\kern{-.8pt}B \over B\kern{-.8pt}C}\) ofwel \(\sin(31\degree)={51 \over B\kern{-.8pt}C}\text{.}\) 1p ○ Hieruit volgt \(B\kern{-.8pt}C={51 \over \sin(31\degree)}\text{.}\) 1p ○ Dus \(B\kern{-.8pt}C≈99{,}0\text{.}\) 1p 3p c Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}Q=55\text{,}\) \(Q\kern{-.8pt}R=76\) en \(\angle P=90\degree\text{.}\) Sinus (3) 007i - Sinus, cosinus en tangens - basis - 0ms c Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle R)={P\kern{-.8pt}Q \over Q\kern{-.8pt}R}\) ofwel \(\sin(\angle R)={55 \over 76}\text{.}\) 1p ○ Hieruit volgt \(\angle R=\sin^{-1}({55 \over 76})\text{.}\) 1p ○ Dus \(\angle R≈46{,}4\degree\text{.}\) 1p 3p d Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}C=73\text{,}\) \(\angle A=53\degree\) en \(\angle B=90\degree\text{.}\) Cosinus (1) 007j - Sinus, cosinus en tangens - basis - 0ms d Cosinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\cos(\angle A)={A\kern{-.8pt}B \over A\kern{-.8pt}C}\) ofwel \(\cos(53\degree)={A\kern{-.8pt}B \over 73}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}B=73⋅\cos(53\degree)\text{.}\) 1p ○ Dus \(A\kern{-.8pt}B≈43{,}9\text{.}\) 1p opgave 23p a Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}Q=52\text{,}\) \(\angle P=32\degree\) 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 \(\cos(32\degree)={52 \over P\kern{-.8pt}R}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R={52 \over \cos(32\degree)}\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈61{,}3\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=28\text{,}\) \(A\kern{-.8pt}B=49\) en \(\angle C=90\degree\text{.}\) Cosinus (3) 007l - Sinus, cosinus en tangens - basis - 0ms b Cosinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\cos(\angle B)={B\kern{-.8pt}C \over A\kern{-.8pt}B}\) ofwel \(\cos(\angle B)={28 \over 49}\text{.}\) 1p ○ Hieruit volgt \(\angle B=\cos^{-1}({28 \over 49})\text{.}\) 1p ○ Dus \(\angle B≈55{,}2\degree\text{.}\) 1p |