Plane trigonometry, by S.L. Loney.

[Exs. L.] SINES AND COSINES OF SMALL ANGLES. 339
37. Find a and b so that the expression a sin x + b sin 2x may be as
close an approximation as possible to the number of radians in the angle
x, when x is small.
38. If y= x - e sin x, where e is very small, prove that
tan = tan (- e+esin2),
and that
tan - tan ( e+e cos2 ),
where powers of e above the second are neglected.
39. If in the equation sin (co - 0) =sin w cos a, 0 be very small, prove
that its approximate value is
2 tan w sin 2 (1- tan2 2 sin2).
2 2 2
40. If 0-be known by means of sin ( to be an angle not > 15', prove
that its value differs from the fraction
28 sin 20 + sin 40
12 (3 + 2 cos 20)
by less than the number of radians in 1'.

22-2