Maîtrisez arcsin et arccos : Quiz interactif pour Terminale Math

Question 1 sur 10 10:00

[{"id":15788,"question":"Quelle est la valeur de arcsin(1\/2) ?","option_a":"π\/6","option_b":"π\/4","option_c":"π\/3","option_d":"π\/2","option_e":"","option_f":"","bonne_reponse":"A","explication":"arcsin(1\/2) = π\/6 car sin(π\/6) = 1\/2 et π\/6 ∈ [-π\/2, π\/2].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15789,"question":"La fonction arccos(x) est-elle définie pour x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux. Le domaine de arccos(x) est [-1, 1], donc x = 2 est hors domaine.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15790,"question":"Quelle est la dérivée de f(x) = arcsin(x²) ?","option_a":"2x\/√(1-x²)","option_b":"x\/√(1-x⁴)","option_c":"2x\/√(1-x⁴)","option_d":"x\/√(1-x²)","option_e":"","option_f":"","bonne_reponse":"C","explication":"En utilisant la règle de la chaîne : f'(x) = (1\/√(1-(x²)²)) * 2x = 2x\/√(1-x⁴).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15791,"question":"Quelle identité relie arcsin(x) et arccos(x) ?","option_a":"arcsin(x) + arccos(x) = 0","option_b":"arcsin(x) + arccos(x) = π\/2","option_c":"arcsin(x) * arccos(x) = π\/2","option_d":"arcsin(x) - arccos(x) = π\/2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour tout x ∈ [-1, 1], arcsin(x) + arccos(x) = π\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15792,"question":"La fonction arcsin(x) est-elle croissante sur son domaine ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai. La dérivée de arcsin(x) est 1\/√(1-x²) \u003E 0 sur ]-1, 1[, donc la fonction est croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15793,"question":"Quelle est l'intégrale de 1\/√(1-x²) ?","option_a":"arcsin(x) + C","option_b":"arccos(x) + C","option_c":"ln|x| + C","option_d":"√(1-x²) + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de 1\/√(1-x²) est arcsin(x) + C, car c'est la dérivée de arcsin(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15794,"question":"Pour x = 0, quelle est la valeur de arccos(x) ?","option_a":"0","option_b":"π\/2","option_c":"π","option_d":"π\/4","option_e":"","option_f":"","bonne_reponse":"B","explication":"arccos(0) = π\/2 car cos(π\/2) = 0 et π\/2 ∈ [0, π].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15795,"question":"La fonction arccos(x) est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux. arccos(-x) = π - arccos(x), donc la fonction n'est pas paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15796,"question":"Quelle est la valeur de sin(arccos(√3\/2)) ?","option_a":"1\/2","option_b":"√3\/2","option_c":"1\/√2","option_d":"0","option_e":"","option_f":"","bonne_reponse":"A","explication":"Soit θ = arccos(√3\/2), alors cos(θ) = √3\/2. Donc sin(θ) = √(1 - (√3\/2)²) = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":15797,"question":"Quelle est la limite de arcsin(x) lorsque x tend vers 1 par la gauche ?","option_a":"0","option_b":"π\/2","option_c":"π","option_d":"1","option_e":"","option_f":"","bonne_reponse":"B","explication":"Lorsque x → 1⁻, arcsin(x) → π\/2 car sin(π\/2) = 1 et arcsin est continue sur [-1, 1].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]