history of mathematics - National STEM Centre

14 Answers
4 If he had a daughter the shares would be 2 : 2 : 2 :1 so
the stranger gets (j-j) + }(}-}) or $, leaving |§ to
be shared by his three sons. The estate is divided into 189
equal parts: the stranger gets 39 and the sons 50 each.
Activity 5.2, page 58
1 a crd 36° =2 sin 18° =0.618 033 989. Con verting
this to sexagesimal, you find that it is 0;37,4,55, the
difference between the two values being 1.58x 10"6 .
b The next sexagesimal figure is 20.
2 Use the expression 2x = crd (2(90 - 50))°.
3 What you need is cos 35° so that you can use the
cosine formula. Use the fact that
cos35°=sin55°=^crdllO°
Activity 5.3, page 58
1 sin30 0 =^-,cos30 0 =^V3, sin45° = cos45°= JrV2,
sin60°=^V3, cos60°= {.
2 a Using isosceles triangles, BC = BD = AD = 1.
Then AB = 2 cos 36° and AC = 1 + 2 cos 72°.
b Writing c — cos36°, you get the quadratic equation
4c 2 - 2c -1 = 0, and you need the positive solution.
c Using the relation cos 2a = 2 cos 2 a - 1 again, you
find that 2cos 2 18°-1 = j(l + Vsj, from which the result
follows.
d Usecos 2 18° + sin 2 18°=l.
3 There are a number of possibilities. They actually
used the fact that sin 72°= cos 18° and cos72°= sin 18°
together with the values of sin 60° and cos 60° in the
formula for sin(72°-60°) .
4 0.017 452 05. This is correct to 6 decimal places.
5 a 0.017 452406. The ninth place doesn't change
after four iterations.
b The convergence is by no means as quick. You need
about 12 iterations to achieve an accurate result.
c Convergence is very slow indeed.
Activity 5.4, page 60
1 He probably summed the GP
1 + 2 + 4 +... + 2 63 = 2 M - 1 and then calculated 2 64 by
repeated squaring.
2 a (14,2), (10,5), (6,8) and (2,11)
170
b Because the solutions cannot be determined precisely.
3 The algorithm is given in the hints section. There are
26 solutions in all.
4 a 204, 577 and 1189, 3363
b The method does apply to 2;c 2 +1 = y2 . Successive
solutions are 2, 3; 12, 17; 70, 99; 408, 577. However, the
method doesn't help to find a starting solution.
Activity 5.5, page 61
1 The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44,
55, 110 which add to 284. The factors of 284 are 1, 2,4,
71, 142 which add to 220.
2 a When n = 2, p = 5,