PROBLEMS 37
PROBLEMS
1. Convert o dgrees to adans 90, 360, -180~, 11~, (180/).
2. Covrt from radians to degres: 74, 3r4, 7, 3r, -, 1 1.7, -7.3.
3. hat is the ar a of se CIf centrl anglea (rians) in a irce of radius r?
4. A re gula polyg o of n sides i inscribed in a irce of radius r.
(a) Find its ara (b) Find its perineter.
5., Ql ate
(a) sin/4) (b) cos(w 3) (c) sin r (d) cos( - /r 6)
(e) tun(7ni ) (f) cs(3i/2) (g) cot( -77r/4)
6. Prove the iduenties:
(a) sx - ) sinx - y I) - sinx - sin2! (b) sin 30 = 3 sin 0 - 4 sin
(c) cos ( )(3 +4 cos 20 1 cos 40) (d) cot 0 = csc 20 + cot 20
7. Solv for: (a) 2 sin sin - 1 = 0 (b) tan 0+ cos = 2.
8. aph the points with the iven polr coordinates
a (3,0) (b) (2, ) (c) (1,57 )
d 2 e '1 22) i(f) (1,1)
9. Find a set of poar coordinates for each of the following points with given Cartesian
coorinates (x, J):
(a) (2,2) ) (-1, 0) (c) (0, -2) d) (3, -2)
10. Find the Cartesan cordinates of the points with given polar coordinates:
(a) (1 ) (b) (3,) (c) (5,27r) (d) (2, )
11. Evaluate:
(a) (3 5i) (2 7i) (b) (1 - i) - (-3) () (1+ i)(1-i)
2 +i (2 + i)
S ((3-)- i (1 + 2i)
(g) cos i sin (h) cos 1 + i sin 1
12. Prove that zl - 2 equals the distance Ibtween the points l, z2:
(a) From the geometri meanig of the addition of complex numbers (Figure 0-29).
(b) By exprssing - 21 in terms of x, yi, 3x, y2
13. Let r(os 0 + i sin 0) 0, how that
(a)1 2 (b) - =-(cos 0 - i sin 0)
rz
14. Proe the rle (0-170).
15. (a) Set n 2 in (0-171) ad take real and imaginar parts on both sides to prove that
cos 20 = cos20 - sin, sin 20 = 2 sin 0 cos 0
(b) Set n = 3 in (0-171) and take real and inaginary pars on both sides to prove
that
cos 30 = cos 0 - 3 cos 0 sin2 0
sin 30 = 3 cos2 0 sin 0 - sin3
16. (a) Prove that Zi. 2 = 0 implies zl = 0 or z2 = 0.
(b) Show that, although for real numbers x2 + y2 = 0 implies x = 0 and y = 0,
for complex numbers zl2 + zz22 = 0 does not imply Zi = 0 or z2 = 0.
