(1) (i) 30o, 150°, -210o, -330o. (ii) 45o, 135o, -225o, - 315o. (iii) 60o, 120o, – 240o, - 300o. (iv) – 30, – 150, 210, 330, 5 п 7π 13π 15п (2) (i) 20o, 160o, 380o, 520o. (ii) 4 4' 4 4

(3) (i) 0=nx+(-1)" ( - ☎). (ii) 0=nw+(−1)"

XXXII. Page 115.

(1) (i) 0=2nπ±}π. (ii) 0=nπ+‡Ã. (iii) 0=nT - IT. (iv) 0=nπ-}π. (v) 0=2nπ±‡π. (vi) 0=nπ+åπ.

(3) The tangent.

(4) No.

(5) (i) 60°, 60°, 300°, -300°. (ii) 45°, 225°, 405°, - 135°. (iii) -45°, -225°, 135o, 315o.

(v) 144, – 144, 216, – 216.

(iv) – 60, – 240, 120°, 300.

(vi) 135, – 450, – 225°, 315®,

(2) sin 24 =, sin 34-√5.

(3) sin 20= √15, sin 30=√15, cos 30= -1; in the second.

(1) (i) +, -. (ii) +. (iii) -, -、 (iv) +,

(vi) +, +. (vii)

(viii)

(3) sin 90=4{V(3+V5) - V(5–15),

cos 90={√(3+√5)+√(5-√5),

sin 810=cos 9o, cos 1890 - cos 9o, tan 2021°=√2-1,

tan 9730 (√3+√2) (√2+1).

(7) Between (2n × 180° - 45o) and (2n x 180o + 45o).

(8) x=-69°26′ 30′′ + 2n × 180° ± (69° 26′ 30′′). (9) x=7504+nx180° +(−1)” (14° 56").

(18) Both equations are satisfied if (x2 - y2)= an odd integer; or, if (x+y)2=2n, and (x − y)2= 4m – 2n, m and n being integers.

(19) (i) 2 sin 0. cos 0=sin 20, and therefore it goes through the same changes as sin 0, while e changes from 0 to 2π. (ii) cos20-sin20 = cos 20. (iii) sin 30 goes through the same changes as sin 0, while goes from 0 to 3. (iv) cot 24; compare with cot A. (v) sin (0+ a) goes through the same changes while goes from 0 to π, as sin goes through while @ goes from a to π+α. (vi) cos (20 - a) goes through the same changes while goes from 0 to π, as cos 20

α

α

goes through as ◊ changes from - to -. (See Ans. to XXVII.) @ -2 π 2

(20) They are the solutions of the equations sin @=sin α, and

(21) They are the solutions of the equations sin (+

= sin

(1) 60206, 9542426, 90309, 7781513, 1-20412, 1.690196. (2) 1.146128, 1.20412, 1.2552726, 1.3802113, 1.4313639, 1.6232493. (3) 1, 69897, 1-1760913, 1.39794, 1-4771213, 1·5440680.

(4) 1.5563026, 1.60206, 1.6812413, 1.69897, 2.30103, 3.

(5) the second decimal place, the first dec. pl., the sixth dec. pl. (6) ten thousands, units, hundreds, third dec. pl., first dec. pl., units. (7) 10, 4, 25, 31. (8) 9, 11, 85, 4, 9, 6.

(9) units, fourth dec. pl., thousands, seventh dec. pl., second dec. pl. (10) tenth integral pl., twelfth dec. pl., fifth dec. pl., units, twelfth dec. pl., first dec. pl.

LV. Pages 171, 172.

(1) 2.8901023, 8901023, 4.8901023, 5.8901023.

(2) 6.7714552, ·7714552, 4·7714552, 2·7714552, 3·7714552. (3) 27724... (4) 00001638... (5) 77448... (6) '005968...

(11) 0,

(12) 63-31-32.

(19) 2.53855.

b+1' 2b+2' b+1' b+1' 2b+2' b+1°

(13) (all-a10) integers. (14) 1.9485 nearly. (20) 4.53999.

LVIII. Pages 181, 182.

(2) 2-7513738. (3) 4.9413333.

(21) 167 years.

(1) ⚫8839066.

(4) 6.8086920.

(5) ⚫5710750.

(6) 3.70404.