Getal & Ruimte (12e editie) - havo wiskunde B
'Sinus, cosinus en tangens'.
| 3 havo | 6.3 Berekeningen met de tangens |
opgave 13p a Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}M = 31 \text{,}\) \(\angle M = 57\degree\) en \(\angle K = 90\degree \text{.}\) Tangens (1) 007m - Sinus, cosinus en tangens - basis - 0ms a Tangens in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\tan(\angle M) = {K\kern{-.8pt}L \over K\kern{-.8pt}M}\) ofwel \(\tan(57\degree) = {K\kern{-.8pt}L \over 31} \text{.}\) 1p ○ Hieruit volgt \(K\kern{-.8pt}L = 31 ⋅ \tan(57\degree) \text{.}\) 1p ○ Dus \(K\kern{-.8pt}L ≈ 47{,}7 \text{.}\) 1p 3p b Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R = 37 \text{,}\) \(\angle Q = 33\degree\) 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 \(\tan(33\degree) = {37 \over Q\kern{-.8pt}R} \text{.}\) 1p ○ Hieruit volgt \(Q\kern{-.8pt}R = {37 \over \tan(33\degree)} \text{.}\) 1p ○ Dus \(Q\kern{-.8pt}R ≈ 57{,}0 \text{.}\) 1p 3p c Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}M = 44 \text{,}\) \(K\kern{-.8pt}L = 21\) en \(\angle K = 90\degree \text{.}\) Tangens (3) 007o - Sinus, cosinus en tangens - basis - 0ms c Tangens in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\tan(\angle M) = {K\kern{-.8pt}L \over K\kern{-.8pt}M}\) ofwel \(\tan(\angle M) = {21 \over 44} \text{.}\) 1p ○ Hieruit volgt \(\angle M = \tan^{-1}({21 \over 44}) \text{.}\) 1p ○ Dus \(\angle M ≈ 25{,}5\degree \text{.}\) 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 \(P\kern{-.8pt}Q = 47 \text{,}\) \(\angle Q = 38\degree\) en \(\angle R = 90\degree \text{.}\) Sinus (1) 007g - Sinus, cosinus en tangens - basis - 0ms a Sinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\sin(\angle Q) = {P\kern{-.8pt}R \over P\kern{-.8pt}Q}\) ofwel \(\sin(38\degree) = {P\kern{-.8pt}R \over 47} \text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R = 47 ⋅ \sin(38\degree) \text{.}\) 1p ○ Dus \(P\kern{-.8pt}R ≈ 28{,}9 \text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C = 53 \text{,}\) \(\angle A = 55\degree\) en \(\angle B = 90\degree \text{.}\) Sinus (2) 007h - Sinus, cosinus en tangens - basis - 0ms b Sinus in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\sin(\angle A) = {B\kern{-.8pt}C \over A\kern{-.8pt}C}\) ofwel \(\sin(55\degree) = {53 \over A\kern{-.8pt}C} \text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}C = {53 \over \sin(55\degree)} \text{.}\) 1p ○ Dus \(A\kern{-.8pt}C ≈ 64{,}7 \text{.}\) 1p 3p c Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(L\kern{-.8pt}M = 53 \text{,}\) \(K\kern{-.8pt}M = 60\) 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 \(\sin(\angle K) = {53 \over 60} \text{.}\) 1p ○ Hieruit volgt \(\angle K = \sin^{-1}({53 \over 60}) \text{.}\) 1p ○ Dus \(\angle K ≈ 62{,}0\degree \text{.}\) 1p 3p d Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}M = 40 \text{,}\) \(\angle K = 52\degree\) en \(\angle L = 90\degree \text{.}\) Cosinus (1) 007j - Sinus, cosinus en tangens - basis - 0ms d Cosinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\cos(\angle K) = {K\kern{-.8pt}L \over K\kern{-.8pt}M}\) ofwel \(\cos(52\degree) = {K\kern{-.8pt}L \over 40} \text{.}\) 1p ○ Hieruit volgt \(K\kern{-.8pt}L = 40 ⋅ \cos(52\degree) \text{.}\) 1p ○ Dus \(K\kern{-.8pt}L ≈ 24{,}6 \text{.}\) 1p opgave 23p a Gegeven is \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) met \(K\kern{-.8pt}L = 46 \text{,}\) \(\angle K = 36\degree\) en \(\angle L = 90\degree \text{.}\) Cosinus (2) 007k - Sinus, cosinus en tangens - basis - 0ms a Cosinus in \(\triangle K\kern{-.8pt}L\kern{-.8pt}M\) geeft \(\cos(\angle K) = {K\kern{-.8pt}L \over K\kern{-.8pt}M}\) ofwel \(\cos(36\degree) = {46 \over K\kern{-.8pt}M} \text{.}\) 1p ○ Hieruit volgt \(K\kern{-.8pt}M = {46 \over \cos(36\degree)} \text{.}\) 1p ○ Dus \(K\kern{-.8pt}M ≈ 56{,}9 \text{.}\) 1p 3p b Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}Q = 23 \text{,}\) \(P\kern{-.8pt}R = 53\) en \(\angle Q = 90\degree \text{.}\) Cosinus (3) 007l - Sinus, cosinus en tangens - basis - 0ms b Cosinus in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\cos(\angle P) = {P\kern{-.8pt}Q \over P\kern{-.8pt}R}\) ofwel \(\cos(\angle P) = {23 \over 53} \text{.}\) 1p ○ Hieruit volgt \(\angle P = \cos^{-1}({23 \over 53}) \text{.}\) 1p ○ Dus \(\angle P ≈ 64{,}3\degree \text{.}\) 1p |