299. By means of the identities

2 sin2 a = 1-cos 2a,

4 sin3 a 3 sin a - sin 3a,

=

2 cos2 a=1+cos 2a,

4 cos3 a 3 cos a+cos 3a,

we can find the sum of the squares and cubes of the sines and cosines of a series of angles in arithmetical progression.

Example 1. Find the sum of n terms of the series

sin2a+sin2 (a+ẞ) + sin2 (a+28) + .......

28= {1-cos 2a} + {1 −cos (2a +2ẞ)} + {1 − cos (2a +4ß) } +

=n- - {cos 2a+cos (2a+28) + cos (2a +48) +

...};

Example 2. Find the sum of the series

cos3 a + cos3 3a + cos3 5a+ +cos3 (2n-1) a.

......

4S=(3 cos a+cos 3a) + (3 cos 3a + cos 9a) + (3 cos 5a + cos 15a) + ......

300. The following further examples illustrate the principle of Art. 293.

Example 2. Find the sum of n terms of the series

Replacing a by and dividing by 2, we obtain

Sum each of the following series to n terms:

1. cos2 8+cos2 30+ cos2 50+.......

1

9. sin2 0 sin 20+ sin2 20 sin 40+ sin2 40 sin 80 +....

1

4

15. From any point on the circumference of a circle of radius r, chords are drawn to the angular points of the regular inscribed polygon of n sides: shew that the sum of the squares of the chords is 2nr2.

16. From a point P within a regular polygon of 2n sides, perpendiculars PA1, PA2, PA3, ...PA are drawn to the sides:

shew that

2n

PA1+PA+...+PAgn-1=PA2+PA4+...+PA=nr,

where r is the radius of the inscribed circle.

17. If  ̧‚ ̧...A2n+1 is a regular polygon and P a point on the circumscribed circle lying on the arc A1A2n+19 shew that PA2+PA+...+PA2n+1=PA2+PA+...+PA2n⋅

18. From any point on the circumference of a circle, perpendiculars are drawn to the sides of the regular circumscribing polygon of n sides: shew that

(1) the sum of the squares of the perpendiculars is

3nr2

;

2

5nr3

(2) the sum of the cubes of the perpendiculars is

2

CHAPTER XXIV.

MISCELLANEOUS TRANSFORMATIONS AND IDENTITIES.

Symmetrical Expressions.

301. An expression is said to be symmetrical with respect to certain of the letters it contains, if the value of the expression remains unaltered when any pair of these letters are interchanged. Thus

cos a+cos B+cos y, sin a sin ẞ sin y,

tan (a-0)+tan (6-0)+tan (y-0),

are expressions which are symmetrical with respect to the letters α, β, γ.

302. A symmetrical expression involving the sum of a number of quantities may be concisely denoted by writing down one of the terms and prefixing the symbol Σ. Thus Σ cos a stands for the sum of all the terms of which cosa is the type, Σ sin asin ẞ stands for the sum of all the terms of which sin a sing is the type; and so on.

For instance, if the expression is symmetrical with respect to the three letters a, B, Y,

cos B cos y = cos ẞ cos y+cos y cos a+ cos a cos ß;

Σ sin (a-0)=sin (a-0)+sin (8-0) + sin (y-0).

303. A symmetrical expression involving the product of a number of quantities may be denoted by writing down one of the factors and prefixing the symbol II. Thus II sin a stands for the product of all the factors of which sin a is the type.

For instance, if the expression is symmetrical with respect to the three letters a, B, y,

II tan (a+8)=tan (a+0) tan (B+0) tan (y+0);

II (cos ẞ+cos y) = (cos B+cos y) (cos y+cos a) (cos a+cos B).

304. With the notation just explained, certain theorems in Chap. XII. involving the three angles A, B, C, which are connected by the relation A+B+C=180°, may be written more concisely. For instance

sin - sin cos - cos sin cos - cos & sin

NOTE. This result is important in Analytical Geometry.

It should be remarked that cos (0-4) is a symmetrical function of 0 and 4, for cos (0-4)=cos (4-0); hence the values obtained for a b c involve 0 and 4 symmetrically.

Example 2. If a and ẞ are two different values of which satisfy the equation a cos 0+b sin 0=c, find the values of