Correction - Exercice 02 page 46 - Rapports trigonométriques d'un angle aigu - Relations métriques dans un triangle rectangle
1ère année secondaire
Rapports trigonométriques d'un angle aigu
Relations métriques dans un triangle rectangle
Exercice 02 page 46
Soit \(STP\) un triangle rectangle en \(S\) et \([SO]\) est une hauteur.
On a :
\(\widehat{P} = 90°-36°=\) \(54°\)
- Cherchons \(ST\)
On a :
\(cos~\widehat{T} = \frac{ST}{TP}\) \(\Rightarrow\)
\(cos~36° = \frac{ST}{12}\) \(\Rightarrow\)
\(cos~\widehat{T} = \frac{ST}{TP}\) \(\Rightarrow\)
\(cos~36° = \frac{ST}{12}\) \(\Rightarrow\)
\(ST = cos~36°\times12\) \(\Rightarrow\)
\(ST = 0,81\times12\) \(\Rightarrow\)
\(ST\approx \) \(9,7\)
- Cherchons \(SP\)
On a :
\(ST^2+SP^2=TP^2\) \(\Rightarrow\)
\(ST^2+SP^2=TP^2\) \(\Rightarrow\)
\(SP^2=TP^2-ST^2\) \(\Rightarrow\)
\(SP^2=12^2-9,7^2\) \(\Rightarrow\)
\(SP^2=144-94,09\) \(\Rightarrow\)
\(SP^2=144-94,09\) \(\Rightarrow\)
\(SP^2=49,91\) \(\Rightarrow\)
\(SP^2=\sqrt{49,91}\) \(\Rightarrow\)
\(SP\approx\) \(7,1\)
- Cherchons \(OS\)
On a :
\(OS = sin~54°\times7,1\) \(\Rightarrow\)
\(OS = 0,81\times7,1\) \(\Rightarrow\)
\(OS\approx \) \(5,8\)
- Cherchons \(OT\)
\(OT = cos~36°\times9,7\) \(\Rightarrow\)
\(OT = 0,81\times9,7\) \(\Rightarrow\)
\(OT\approx \) \(7,9\)
- Cherchons \(OP\)
- Cherchons \(\widehat{T}\)
On a :
\(\widehat{T} = 90°-25°=\) \(65°\)
- Cherchons \(SP\)
\(SP = \frac{7}{tg~25°}\) \(\Rightarrow\)
\(SP = \frac{7}{0,47}\) \(\Rightarrow\)
\(SP= \) \(14,9\)
- Cherchons \(TP\)
- Cherchons \(OS\)
On a :
\(OS = sin~25°\times14,9\) \(\Rightarrow\)
\(OS = 0,42\times14,9\) \(\Rightarrow\)
\(OS\approx \) \(6,3\)
- Cherchons \(OT\)
\(OT = \frac{cos~65°}{7}\)\(\Rightarrow\)
\(OT = 0,42\times7\) \(\Rightarrow\)
\(sin~\widehat{P}=\frac{OS}{SP}\) \(\Rightarrow\)
\(sin~54° = \frac{OS}{7,1}\) \(\Rightarrow\)
\(OS = sin~54°\times7,1\) \(\Rightarrow\)
\(OS = 0,81\times7,1\) \(\Rightarrow\)
\(OS\approx \) \(5,8\)
- Cherchons \(OT\)
On a :
\(cos~\widehat{T} = \frac{OT}{ST}\) \(\Rightarrow\)
\(cos~36° = \frac{OT}{9,7}\) \(\Rightarrow\)
\(cos~\widehat{T} = \frac{OT}{ST}\) \(\Rightarrow\)
\(cos~36° = \frac{OT}{9,7}\) \(\Rightarrow\)
\(OT = cos~36°\times9,7\) \(\Rightarrow\)
\(OT = 0,81\times9,7\) \(\Rightarrow\)
\(OT\approx \) \(7,9\)
- Cherchons \(OP\)
On a :
\(OP = TP-OT\) \(\Rightarrow\)
\(OP = 12-7,9\) \(\Rightarrow\)
\(OP= \) \(4,1\)
\(OP = TP-OT\) \(\Rightarrow\)
\(OP = 12-7,9\) \(\Rightarrow\)
\(OP= \) \(4,1\)
* 2ème ligne :
On a :
\(\widehat{T} = 90°-25°=\) \(65°\)
- Cherchons \(SP\)
On a :
\(tg~\widehat{P} = \frac{ST}{SP}\) \(\Rightarrow\)
\(tg~25° = \frac{7}{SP}\) \(\Rightarrow\)
\(tg~\widehat{P} = \frac{ST}{SP}\) \(\Rightarrow\)
\(tg~25° = \frac{7}{SP}\) \(\Rightarrow\)
\(SP = \frac{7}{tg~25°}\) \(\Rightarrow\)
\(SP = \frac{7}{0,47}\) \(\Rightarrow\)
\(SP= \) \(14,9\)
- Cherchons \(TP\)
On a :
\(TP^2=ST^2+SP^2\)\(\Rightarrow\)
\(TP^2=ST^2+SP^2\)\(\Rightarrow\)
\(TP^2=7^2+14,9^2\)\(\Rightarrow\)
\(TP^2=49+222,01\)\(\Rightarrow\)
\(TP^2=271,01\)\(\Rightarrow\)
\(TP^2=\sqrt{271,01}\)\(\Rightarrow\)
\(TP\approx\) \(16,5\)
\(TP^2=271,01\)\(\Rightarrow\)
\(TP^2=\sqrt{271,01}\)\(\Rightarrow\)
\(TP\approx\) \(16,5\)
- Cherchons \(OS\)
On a :
\(sin~\widehat{P}=\frac{OS}{SP}\) \(\Rightarrow\)
\(sin~25° = \frac{OS}{14,9}\) \(\Rightarrow\)
\(OS = sin~25°\times14,9\) \(\Rightarrow\)
\(OS = 0,42\times14,9\) \(\Rightarrow\)
\(OS\approx \) \(6,3\)
- Cherchons \(OT\)
On a :
\(cos~\widehat{T} = \frac{OT}{ST}\) \(\Rightarrow\)
\(cos~65° = \frac{OT}{7}\) \(\Rightarrow\)
\(cos~\widehat{T} = \frac{OT}{ST}\) \(\Rightarrow\)
\(cos~65° = \frac{OT}{7}\) \(\Rightarrow\)
\(OT = \frac{cos~65°}{7}\)\(\Rightarrow\)
\(OT = 0,42\times7\) \(\Rightarrow\)
On a :
\(\widehat{P} = 90°-50°=\) \(40°\)
- Cherchons \(OT\)
On a :
\(tg~\widehat{T} = \frac{OS}{OT}\) \(\Rightarrow\)
\(tg~50° = \frac{9,5}{OT}\) \(\Rightarrow\)
\(tg~\widehat{T} = \frac{OS}{OT}\) \(\Rightarrow\)
\(tg~50° = \frac{9,5}{OT}\) \(\Rightarrow\)
\(OT = \frac{9,5}{tg~50°}\)\(\Rightarrow\)
\(OT = \frac{9,5}{1,19}\) \(\Rightarrow\)
- Cherchons \(ST\)
On a :
\(sin~\widehat{T} = \frac{OS}{ST}\) \(\Rightarrow\)
\(sin~50° = \frac{9,5}{ST}\) \(\Rightarrow\)
\(sin~\widehat{T} = \frac{OS}{ST}\) \(\Rightarrow\)
\(sin~50° = \frac{9,5}{ST}\) \(\Rightarrow\)
\(ST = \frac{9,5}{sin~50°}\) \(\Rightarrow\)
\(OT = \frac{9,5}{0,77}\) \(\Rightarrow\)
\(ST\approx \) \(12,3\)
- Cherchons \(SP\)
On a :
\(SP = \frac{9,5}{sin~40°}\)\(\Rightarrow\)
\(SP = \frac{9,5}{0,64}\)\(\Rightarrow\)
\(SP\approx \) \(14,8\)
\(sin~\widehat{P}=\frac{OS}{SP}\) \(\Rightarrow\)
\(sin~40° = \frac{9,5}{SP}\) \(\Rightarrow\)
\(SP = \frac{9,5}{sin~40°}\)\(\Rightarrow\)
\(SP = \frac{9,5}{0,64}\)\(\Rightarrow\)
\(SP\approx \) \(14,8\)
- Cherchons \(TP\)
On a :
\(TP^2=151,29+219,04\) \(\Rightarrow\)
\(TP^2=370,33\) \(\Rightarrow\)
\(TP^2=\sqrt{370,33}\) \(\Rightarrow\)
\(TP\approx \) \(19,2\)
- Cherchons \(OP\)
\(TP^2=ST^2+SP^2\) \(\Rightarrow\)
\(TP^2=12,3^2+14,8^2\) \(\Rightarrow\)
\(TP^2=151,29+219,04\) \(\Rightarrow\)
\(TP^2=370,33\) \(\Rightarrow\)
\(TP^2=\sqrt{370,33}\) \(\Rightarrow\)
\(TP\approx \) \(19,2\)
- Cherchons \(OP\)
On a :
\(OP = TP-OT\) \(\Rightarrow\)
\(OP = 19,2-8\) \(\Rightarrow\)
\(OP= \) \(11,2\)
\(OP = TP-OT\) \(\Rightarrow\)
\(OP = 19,2-8\) \(\Rightarrow\)
\(OP= \) \(11,2\)
Libellés:
1ère année secondaire
Correction
Corrigées
exercice
Le Mathématicien
manuel scolaire
Math
Mathématiques
Rapports trigonométriques d'un angle aigu
Relations métriques dans un triangle rectangle
Aucun commentaire: