Ex. 2. Find the value, when x is zero, of the expression

The value of the expression may be also found by finding the value of

its logarithm.

prove that is the number of radians in 4° 24′ nearly.

find an approximate value for 0.

Find the value, when x is zero, of the expressions

and deduce that in a triangle ABC, in which C is a right angle and CA is

five times CB, the angle A exceeds the eighth part of a right angle by 3' 36", correct to the nearest second.

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

where powers of e above the second are neglected.

39. If in the equation sin (w-0)=sin w cos a, ◊ be very small, prove that its approximate value is

40. If be known by means of sin o to be an angle not > 15', prove that its value differs from the fraction

28 sin 20+ sin 4ø

12 (3+2 cos 2p)

by less than the number of radians in 1'.

CHAPTER XXIV.

EXPANSIONS OF SINES AND COSINES OF MULTIPLE ANGLES, AND OF POWERS OF SINES AND COSINES.

[On a first reading of the subject the student is recommended to omit from the beginning of Art. 293 to the end of the chapter.]

287. In this chapter we shall shew how to expand powers of cosines and sines of an angle in terms of cosines and sines of multiples of that angle, and also how to express cosines and sines of multiple angles in terms of powers of cosines and sines.

Also, by De Moivre's Theorem, we have

x2 = cos no + i sin no,