Quiz : Fonctions trigonométriques et équations — 1ère année secondaire

Question 1 sur 10 10:00

[{"id":5635,"question":"Quelle est la valeur de sin(π\/6) ?","option_a":"1\/2","option_b":"√2\/2","option_c":"√3\/2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La valeur de sin(π\/6) est 1\/2, c'est une valeur fondamentale à retenir en trigonométrie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5636,"question":"L'équation sin(x) = 0 a pour solutions x = kπ, où k est un entier.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est vrai : sin(x) = 0 si et seulement si x = kπ, avec k ∈ ℤ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5637,"question":"Quelle est la dérivée de la fonction f(x) = cos(2x) ?","option_a":"-2sin(2x)","option_b":"-sin(2x)","option_c":"2sin(2x)","option_d":"sin(2x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de cos(u) est -u' sin(u). Ici u = 2x, donc u' = 2. D'où f'(x) = -2sin(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5638,"question":"L'équation cos(x) = -1\/2 admet 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":"Vrai : cos(x) = -1\/2 admet deux solutions dans [0, 2π], à savoir 2π\/3 et 4π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5639,"question":"Quelle est la période de la fonction f(x) = sin(3x) ?","option_a":"2π\/3","option_b":"π\/3","option_c":"3π","option_d":"6π","option_e":"","option_f":"","bonne_reponse":"B","explication":"La période de sin(kx) est 2π\/k. Ici k = 3, donc la période est 2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5640,"question":"L'équation sin(x) = cos(x) peut s'écrire tan(x) = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai : sin(x) = cos(x) ⇒ tan(x) = sin(x)\/cos(x) = 1 (si cos(x) ≠ 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5641,"question":"Quelle est la valeur de cos(π\/4) ?","option_a":"1\/2","option_b":"√2\/2","option_c":"√3\/2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"B","explication":"La valeur de cos(π\/4) est √2\/2, une valeur à connaître par cœur en trigonométrie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5642,"question":"L'équation sin²(x) + cos²(x) = 1 est toujours vraie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai : c'est l'identité trigonométrique fondamentale, valable pour tout x réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5643,"question":"Quelle est la dérivée de la fonction f(x) = tan(x) ?","option_a":"1 + tan²(x)","option_b":"1 - tan²(x)","option_c":"sec²(x)","option_d":"csc²(x)","option_e":"","option_f":"","bonne_reponse":"C","explication":"La dérivée de tan(x) est 1 + tan²(x), qui est aussi égale à sec²(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5644,"question":"L'équation cos(2x) = 0 admet 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":"Vrai : cos(2x) = 0 ⇒ 2x = π\/2 + kπ ⇒ x = π\/4 + kπ\/2. Dans [0, π], les solutions sont π\/4 et 3π\/4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]