Plane trigonometry, by S.L. Loney.

358

TRIGONOMETRY.

Equation (1) then becomes
64 cos7 8 - 112 cos5 0 + 56 c3-7os3 7 cos + 1 =0...........(2),
and its roots are
7r 37r 57r 7 7r 9r 11wr 137r
cos C, c os -, c os -, c os, cos, cos and cos -
Now
7r 137r 7r ll 3r 97r 57r
cos - -1, cos = -=coos -, co s, and cos -=cos.
7 c05- 7 7 7
r 3wr 5w
The roots of (2) are therefore -1, and cos -, cos -, and cos -, the
7' 7 7
latter three roots being twice repeated.
Writing c, for shortness, for cos 0, the equation (2) may be written
(c + 1) (8c — 4c2-4c + 1)2=0.
7r 3 57r
Hence cos -, cos -, and cos s are the roots of the equation
8c -4c -4c2-4c+1=0......................... (3).
We therefore have
7r 37r 57r 4 1
cos + cos -+ cos - = =
COS7+C 7 +cos 7 =8=2'
7r 37r 37r 57r 57r 7r -4 1
cos V cos - + cos - cos - + cos-y cos -g- -
7 37r 57r - 1 1
and cosCos- cos 7 = - 8- -
I 1
In equation (3), putting -=x, and therefore c= ~- it follows that the
quantities sec2 7 sec2, and sec2 are the roots of the equation
8. 1 4 4
8. / --- +1=0,
xiX x Jx
or, on rationalizing,
x3 - 24x2+80 - 64=0......................... (4).
Again, putting x = 1 +y, then, since sec2 0 = + tan2 0, it follows that
tan2a, tan2 3, and tan2 5
' ' 7
are the roots of the equation
(1+y)3 - 24 (1 y)2 + 80 (1+y) - 64= —0,
i.e. of y3- 2ly2 + 35y - 7 = 0.