2
Sinus, kosinus, tangens va ishoralar
maktab darsligi: algebra 9-sinf, sin, cos, tg, ctg ta’riflari
Maqsad: Birlik aylana yordamida sin, cos, tg ni aniqlash, jadval qiymatlarini va chorakdagi ishoralarni bilish.
Qoida va g‘oya
Birlik aylanada (radiusi 1, markazi koordinata boshida) α burchakka mos nuqta P(cos α; sin α): abssissa — kosinus, ordinata — sinus. tg α = sin α/cos α (cos α ≠ 0), ctg α = cos α/sin α (sin α ≠ 0). Jadval: sin 0° = 0, sin 30° = 1/2, sin 45° = sqrt(2)/2, sin 60° = sqrt(3)/2, sin 90° = 1; kosinuslar shu qiymatlarning teskari tartibi: cos 0° = 1, cos 30° = sqrt(3)/2, cos 60° = 1/2, cos 90° = 0. tg 45° = 1, tg 30° = sqrt(3)/3, tg 60° = sqrt(3). Ishoralar: I chorak — hammasi musbat; II — faqat sin; III — faqat tg va ctg; IV — faqat cos. Qiymatlar: -1 ≤ sin α ≤ 1, -1 ≤ cos α ≤ 1.
Namunaviy masalalar
cos π va sin(3π/2) ni toping.
Yechim: π nuqta (-1; 0), 3π/2 nuqta (0; -1). Demak cos π = -1, sin(3π/2) = -1.
Javob: -1 va -1
Yechim: π nuqta (-1; 0), 3π/2 nuqta (0; -1). Demak cos π = -1, sin(3π/2) = -1.
Javob: -1 va -1
sin 200°, cos 300° va tg 100° ishoralarini aniqlang.
Yechim: 200° — III chorak: sin < 0. 300° — IV chorak: cos > 0. 100° — II chorak: tg < 0.
Javob: manfiy, musbat, manfiy
Yechim: 200° — III chorak: sin < 0. 300° — IV chorak: cos > 0. 100° — II chorak: tg < 0.
Javob: manfiy, musbat, manfiy
tg 60° · ctg 60° ni hisoblang.
Yechim: Ctg tg ning teskarisi, ko‘paytma 1.
Javob: 1
Yechim: Ctg tg ning teskarisi, ko‘paytma 1.
Javob: 1
Tipik xatolar
- Kosinus va sinus qiymatlarini jadvalda almashtirib yuborish: cos 60° = 1/2, sin 60° = sqrt(3)/2.
- III chorakda sinus va kosinus ikkalasi manfiy; tangens musbat.
- tg 90° mavjud emas, uni 0 yoki 1 deb olish.
Mashq
1
sin 30° + cos 60° ni hisoblang.
1
2
cos 140° ning ishorasini aniqlang.
Manfiy
3
sin 90° + cos 180° ni hisoblang.
0
4
tg 45° + ctg 45° ni hisoblang.
2