6) а) sin 245. 180 < 245 < 270 - это 3 четверть, sin 245 < 0
б) ctg 5pi/9 = ctg 10pi/18. 9pi/18 < 10pi/18 < 18pi/18 - это 2 четверть, ctg 5pi/9 < 0
7) а) cos 1140 + tg(-495) = cos(1080+60) - tg(540-45) = cos 60 + tg 45 = 1/2 + 1 = 3/2
б) cos(7pi/10)*cos(4pi/5) - sin(7pi/10)*sin(4pi/5) =
= cos(7pi/10)*cos(8pi/10) - sin(7pi/10)*sin(8pi/10) = cos (7pi/10 + 8pi/10) =
= cos (15pi/10) = cos (3pi/2) = 0
8) pi/2 < a < pi, то есть 2 четверть. sin a > 0, cos a < 0.
tg a = -9/40
sin a = -tg a / √(1 + tg^2 a) = (9/40) / √(1 + 81/1600) = (9/40) / (41/40) = 9/41
cos a = -1 / √(1 + tg^2 a) = -1 / √(1 + 81/1600) = -1 / (41/40) = 40/41
ctg a = 1/tg a = -40/9
9) а) (1 - sin 2a) / (sin a - cos a)^2 =
= (sin^2 a + cos^2 a - 2sin a*cos a) / (sin^2 a - 2sin a*cos a + cos^2 a) = 1
б) [cos (2pi+a) - cos (3pi/2+a)] / [sin (pi/2+2a) + 2sin^2 (pi-a) + sin (pi+2a)] =
= (cos a - sin a) / (cos 2a + 2sin^2 a - sin a) =
= (cos a - sin a) / (1 - 2sin^2 a + 2sin^2 a - sin a) = (cos a - sin a) / (1 - sin a)
10) tg a + ctg a = sin a/cos a + cos a/sin a = 3
(sin^2 a + cos^2 a) / (sin a*cos a) = 1 / (sin a*cos a) = 3
sin a*cos a = 1/3
(sin a+cos a)^2 = sin^2 a + cos^2 a + 2sin a*cos a = 1 + 2sin a*cos a = 1 + 2/3 = 5/3
sin a + cos a = √(5/3)