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 \(A\kern{-.8pt}C=30\text{,}\) \(\angle C=43\degree\) en \(\angle A=90\degree\text{.}\) Tangens (1) 007m - Sinus, cosinus en tangens - basis - 0ms a Tangens in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\tan(\angle C)={A\kern{-.8pt}B \over A\kern{-.8pt}C}\) ofwel \(\tan(43\degree)={A\kern{-.8pt}B \over 30}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}B=30⋅\tan(43\degree)\text{.}\) 1p ○ Dus \(A\kern{-.8pt}B≈28{,}0\text{.}\) 1p 3p b Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}M=27\text{,}\) \(\angle L=56\degree\) en \(\angle M=90\degree\text{.}\) Tangens (2) 007n - Sinus, cosinus en tangens - basis - 0ms b Tangens in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\tan(\angle L)={K\kern{-.8pt}M \over L\kern{-.8pt}M}\) ofwel \(\tan(56\degree)={27 \over L\kern{-.8pt}M}\text{.}\) 1p ○ Hieruit volgt \(L\kern{-.8pt}M={27 \over \tan(56\degree)}\text{.}\) 1p ○ Dus \(L\kern{-.8pt}M≈18{,}2\text{.}\) 1p 3p c Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(L\kern{-.8pt}M=49\text{,}\) \(K\kern{-.8pt}M=36\) en \(\angle M=90\degree\text{.}\) Tangens (3) 007o - Sinus, cosinus en tangens - basis - 0ms c Tangens in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\tan(\angle L)={K\kern{-.8pt}M \over L\kern{-.8pt}M}\) ofwel \(\tan(\angle L)={36 \over 49}\text{.}\) 1p ○ Hieruit volgt \(\angle L=\tan^{-1}({36 \over 49})\text{.}\) 1p ○ Dus \(\angle L≈36{,}3\degree\text{.}\) 1p |
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| 3 vwo | 6.4 De sinus en de cosinus |
opgave 13p a Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(Q\kern{-.8pt}R=55\text{,}\) \(\angle R=51\degree\) en \(\angle P=90\degree\text{.}\) Sinus (1) 007g - Sinus, cosinus en tangens - basis - 0ms a Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle R)={P\kern{-.8pt}Q \over Q\kern{-.8pt}R}\) ofwel \(\sin(51\degree)={P\kern{-.8pt}Q \over 55}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}Q=55⋅\sin(51\degree)\text{.}\) 1p ○ Dus \(P\kern{-.8pt}Q≈42{,}7\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}C=28\text{,}\) \(\angle B=46\degree\) en \(\angle C=90\degree\text{.}\) Sinus (2) 007h - Sinus, cosinus en tangens - basis - 0ms b Sinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\sin(\angle B)={A\kern{-.8pt}C \over A\kern{-.8pt}B}\) ofwel \(\sin(46\degree)={28 \over A\kern{-.8pt}B}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}B={28 \over \sin(46\degree)}\text{.}\) 1p ○ Dus \(A\kern{-.8pt}B≈38{,}9\text{.}\) 1p 3p c Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=33\text{,}\) \(A\kern{-.8pt}C=45\) en \(\angle B=90\degree\text{.}\) Sinus (3) 007i - Sinus, cosinus en tangens - basis - 0ms c Sinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\sin(\angle A)={B\kern{-.8pt}C \over A\kern{-.8pt}C}\) ofwel \(\sin(\angle A)={33 \over 45}\text{.}\) 1p ○ Hieruit volgt \(\angle A=\sin^{-1}({33 \over 45})\text{.}\) 1p ○ Dus \(\angle A≈47{,}2\degree\text{.}\) 1p 3p d Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(Q\kern{-.8pt}R=68\text{,}\) \(\angle R=49\degree\) en \(\angle P=90\degree\text{.}\) Cosinus (1) 007j - Sinus, cosinus en tangens - basis - 0ms d Cosinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\cos(\angle R)={P\kern{-.8pt}R \over Q\kern{-.8pt}R}\) ofwel \(\cos(49\degree)={P\kern{-.8pt}R \over 68}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R=68⋅\cos(49\degree)\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈44{,}6\text{.}\) 1p opgave 23p a Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}B=44\text{,}\) \(\angle A=53\degree\) en \(\angle B=90\degree\text{.}\) Cosinus (2) 007k - Sinus, cosinus en tangens - basis - 0ms a Cosinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\cos(\angle A)={A\kern{-.8pt}B \over A\kern{-.8pt}C}\) ofwel \(\cos(53\degree)={44 \over A\kern{-.8pt}C}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}C={44 \over \cos(53\degree)}\text{.}\) 1p ○ Dus \(A\kern{-.8pt}C≈73{,}1\text{.}\) 1p 3p b Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}M=39\text{,}\) \(L\kern{-.8pt}M=47\) en \(\angle K=90\degree\text{.}\) Cosinus (3) 007l - Sinus, cosinus en tangens - basis - 0ms b Cosinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\cos(\angle M)={K\kern{-.8pt}M \over L\kern{-.8pt}M}\) ofwel \(\cos(\angle M)={39 \over 47}\text{.}\) 1p ○ Hieruit volgt \(\angle M=\cos^{-1}({39 \over 47})\text{.}\) 1p ○ Dus \(\angle M≈33{,}9\degree\text{.}\) 1p |