Question

Oblicz bez użycia tablic:
a)(sin25°+cos25°)(sin25°-cos25°) + 2sin²65°
b)cos²10°+cos²80°
c)(cos17² +cos 73°)² + (sin17°-sin73°)²
d)ctg10° × ctg20° × ctg30°× ctg40° × ctg50° x ctg60° x ctg 70° x ctg80°

Pilne PILNE PIlne

Answer 1

A)
(sin 25⁰ + cos 25⁰)(sin 25⁰ - cos 25⁰) + 2 sin² 65⁰ =
= sin²25⁰ - cos²25⁰ + 2 sin²(90⁰ - 25⁰) =
= sin²25⁰ - cos²25⁰ + 2 * cos²25⁰ = sin²25⁰ + cos²25⁰ = 1
b)
cos²10⁰ + cos²80⁰ = cos²(90⁰ - 80⁰) + cos²80⁰ =
= sin²80⁰ + cos²80⁰ = 1
c)
(cos 17⁰ + cos 73⁰)² + ( sin 17⁰ - sin 73⁰) ² =
= cos²17⁰ + 2 sin 17⁰ * cos 73⁰ + cos²73⁰ +
+ sin²17⁰ - 2 sin 17⁰*sin 73⁰ + sin² 73⁰ =
= (sin²17⁰ + cos²17⁰) + (sin²73⁰ + cos²73⁰) + 2 sin17⁰cos17⁰ -
- 2 sin 17⁰ cos17⁰ = 1 = 1 = 2

bo cos 73⁰ = cos(90⁰ - 17⁰) = sin 17⁰
sin 73⁰ = sin(90⁰ - 17⁰) = cos 17⁰
d)
ctg 80⁰ = ctg(90⁰ - 10⁰) = tg 10⁰
ctg 70⁰ = ctg(90⁰- 20⁰) = tg 20⁰
analogicznie
ctg 60⁰ = tg 30⁰
ctg 50⁰ = tg 40⁰
zatem
ctg 10⁰ *ctg20⁰ *ctg30⁰ *ctg40⁰ * ctg50⁰*ctg60⁰*ctg70⁰*ctg80⁰ =
= ctg10⁰ *ctg80⁰*ctg20⁰*ctg70⁰*ctg30⁰*ctg60⁰*ctg40⁰*ctg50⁰ =
= ctg10⁰*tg10⁰*ctg20⁰*tg20⁰*ctg30⁰*tg30⁰*ctg40⁰*tg40⁰ =
= 1 * 1* 1 * 1 = 1
bo ctg α * tg α = 1
-------------------------------------------------------------------------