Getal & Ruimte (13e editie) - 3 havo
'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 \(P\kern{-.8pt}Q=38\text{,}\) \(\angle P=32\degree\) en \(\angle Q=90\degree\text{.}\) Tangens (1) 007m - Sinus, cosinus en tangens - basis - 0ms a Tangens in \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) geeft \(\tan(\angle P)={Q\kern{-.8pt}R \over P\kern{-.8pt}Q}\) ofwel \(\tan(32\degree)={Q\kern{-.8pt}R \over 38}\text{.}\) 1p ○ Hieruit volgt \(Q\kern{-.8pt}R=38⋅\tan(32\degree)\text{.}\) 1p ○ Dus \(Q\kern{-.8pt}R≈23{,}7\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=44\text{,}\) \(\angle A=31\degree\) 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 \(\tan(31\degree)={44 \over A\kern{-.8pt}B}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}B={44 \over \tan(31\degree)}\text{.}\) 1p ○ Dus \(A\kern{-.8pt}B≈73{,}2\text{.}\) 1p 3p c Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}B=39\text{,}\) \(B\kern{-.8pt}C=44\) en \(\angle B=90\degree\text{.}\) Tangens (3) 007o - Sinus, cosinus en tangens - basis - 0ms c Tangens in \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \(\tan(\angle A)={B\kern{-.8pt}C \over A\kern{-.8pt}B}\) ofwel \(\tan(\angle A)={44 \over 39}\text{.}\) 1p ○ Hieruit volgt \(\angle A=\tan^{-1}({44 \over 39})\text{.}\) 1p ○ Dus \(\angle A≈48{,}4\degree\text{.}\) 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 \(P\kern{-.8pt}Q=49\text{,}\) \(\angle Q=53\degree\) 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 \(\sin(53\degree)={P\kern{-.8pt}R \over 49}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R=49⋅\sin(53\degree)\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈39{,}1\text{.}\) 1p 3p b Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(Q\kern{-.8pt}R=49\text{,}\) \(\angle P=37\degree\) 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 \(\sin(37\degree)={49 \over P\kern{-.8pt}R}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}R={49 \over \sin(37\degree)}\text{.}\) 1p ○ Dus \(P\kern{-.8pt}R≈81{,}4\text{.}\) 1p 3p c Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}Q=33\text{,}\) \(Q\kern{-.8pt}R=52\) 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 \(\sin(\angle R)={33 \over 52}\text{.}\) 1p ○ Hieruit volgt \(\angle R=\sin^{-1}({33 \over 52})\text{.}\) 1p ○ Dus \(\angle R≈39{,}4\degree\text{.}\) 1p 3p d Gegeven is \(\triangle P\kern{-.8pt}Q\kern{-.8pt}R\) met \(P\kern{-.8pt}R=48\text{,}\) \(\angle P=51\degree\) 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 \(\cos(51\degree)={P\kern{-.8pt}Q \over 48}\text{.}\) 1p ○ Hieruit volgt \(P\kern{-.8pt}Q=48⋅\cos(51\degree)\text{.}\) 1p ○ Dus \(P\kern{-.8pt}Q≈30{,}2\text{.}\) 1p opgave 23p a Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}B=25\text{,}\) \(\angle A=55\degree\) 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 \(\cos(55\degree)={25 \over A\kern{-.8pt}C}\text{.}\) 1p ○ Hieruit volgt \(A\kern{-.8pt}C={25 \over \cos(55\degree)}\text{.}\) 1p ○ Dus \(A\kern{-.8pt}C≈43{,}6\text{.}\) 1p 3p b Gegeven is \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C=52\text{,}\) \(A\kern{-.8pt}B=56\) 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 \(\cos(\angle B)={52 \over 56}\text{.}\) 1p ○ Hieruit volgt \(\angle B=\cos^{-1}({52 \over 56})\text{.}\) 1p ○ Dus \(\angle B≈21{,}8\degree\text{.}\) 1p |