Exercises 18 (Polar coordinates)

1. Express in polar form:

   1. $(5, 2)$
   2. $(3, 7)$
   3. $(-5, 2)$
   4. $(-8, -9)$
   5. $(17, -12)$

2. Express in rectangular form:

   1. $5 \angle 65^\circ$
   2. $3.8 \angle 124^\circ$
   3. $7.2 \angle -56^\circ$
   4. $15 \angle -138^\circ$

3. Six holes are marked out in polar coordinates as:

Hole 1	$65 \angle 65^\circ$
Hole 2	$55 \angle 95^\circ$
Hole 3	$45 \angle -20^\circ$
Hole 4	$80 \angle 55^\circ$
Hole 5	$160 \angle -170^\circ$
Hole 6	$95 \angle -80^\circ$

Sketch the system and determine the position of Hole 1 relative to Hole 6 in rectangular coordinates. The coordinates for each hole are given relative to the previous hole.

4. For values of $\theta$ in the range $0$ to $2\pi$, plot at intervals of $\pi/6$ radians the functions:

   1. $r = 2 \sin \theta$
   2. $r = 2 \cos^2\theta$
   3. $r = a(1 + 2 \cos \theta)$
   4. $r = a \cos \theta$
   5. $r = a \sin^2\theta$
   6. $r = a \sin 2\theta$
   7. $r = a \sin 3\theta$
   8. $r = a \cos 2\theta$
   9. $r = a \cos 3\theta$

Assume $a$ is a constant.

Solutions: 1. (i) $5.38\angle21.8^\circ$; (ii) $7.61\angle66.8^\circ$; (iii) $5.38\angle158.2^\circ$; (iv) $12.04\angle-131.6^\circ$; (v) $20.8\angle -35.23^\circ$; 2. (i) $(2.1,4.3)$; (ii) $(-2.1,3.15)$; (iii) $(4.03,-5.97)$; (iv) $(-11.1,-10.0)$; 3. $(57.7,16.4)$; 4.