Quiz : Maîtrisez les fonctions trigonométriques en 4ème année secondaire

Question 1 sur 10 10:00

[{"id":9583,"question":"Quelle est la période de la fonction f(x) = sin(2x) ?","option_a":"π","option_b":"2π","option_c":"π\/2","option_d":"4π","option_e":"","option_f":"","bonne_reponse":"C","explication":"La période d'une fonction sin(kx) est 2π\/k. Ici, k=2, donc la période est 2π\/2 = π.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9584,"question":"L'équation sin(x) = 0,5 admet-elle des solutions dans l'intervalle [0, π] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation sin(x) = 0,5 admet deux solutions dans [0, 2π] : π\/6 et 5π\/6. Donc elle admet au moins une solution dans [0, π].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9585,"question":"Quelle identité trigonométrique est correcte parmi les suivantes ?","option_a":"sin²x + cos²x = 1","option_b":"sin²x - cos²x = 1","option_c":"tan²x + 1 = cot²x","option_d":"1 + cot²x = sin²x","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'identité fondamentale en trigonométrie est sin²x + cos²x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9586,"question":"La fonction f(x) = cos(x) est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une fonction f est paire si f(-x) = f(x). Or cos(-x) = cos(x), donc la fonction cosinus est paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9587,"question":"Quelle est la valeur de tan(π\/4) ?","option_a":"0","option_b":"1","option_c":"√2","option_d":"√3\/2","option_e":"","option_f":"","bonne_reponse":"B","explication":"tan(π\/4) = sin(π\/4)\/cos(π\/4) = (√2\/2)\/(√2\/2) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9588,"question":"L'équation 2cos(x) - 1 = 0 admet-elle des solutions dans l'intervalle [0, 2π] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"2cos(x) - 1 = 0 ⇒ cos(x) = 0,5. Les solutions dans [0, 2π] sont π\/3 et 5π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9589,"question":"Quelle est la dérivée de la fonction f(x) = sin(3x) ?","option_a":"3cos(3x)","option_b":"cos(3x)","option_c":"3sin(3x)","option_d":"-3cos(3x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de sin(u) est u'cos(u). Ici, u=3x ⇒ u'=3, donc f'(x) = 3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9590,"question":"La fonction f(x) = tan(x) est-elle définie pour tout x réel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction tangente n'est pas définie pour x = π\/2 + kπ (k entier), car cos(x) = 0 en ces points.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9591,"question":"Quelle est la valeur de sin(π\/6) ?","option_a":"0","option_b":"1\/2","option_c":"√2\/2","option_d":"√3\/2","option_e":"","option_f":"","bonne_reponse":"B","explication":"sin(π\/6) = 1\/2, valeur à connaître par cœur.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9592,"question":"L'équation sin(x) = cos(x) admet-elle des solutions dans l'intervalle [0, π\/2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"sin(x) = cos(x) ⇒ tan(x) = 1 ⇒ x = π\/4, qui appartient à [0, π\/2].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]