Plane trigonometry, by S.L. Loney.

ELIMINATION.

291

Multiplying (1) by cos 0, (2) by sin 0, and adding, we have
x
- = cos 0 cos 20 + 2 sin 0 sin 20
a
= cos 0 + sin 0 sin 20 = cos 0 +2 sin2 0cos0...............(3).
Multiplying (2) by cos 0, (1) by sin 0, and subtracting, we have
Y=2 sin 20 cos 0 - cos 20 sin 0
= sin 20 cos 0 + sin 0 = sin 0 + 2 sin 0 cos2 0...............(4).
Adding (3) and (4), we have
+- + = (sin 0 + cos 0) [1 + 2 sin 0 cos 0]
ab
= (sin 0 + cos 0) [sin2 0 + cos2 0 + 2 sin 0 cos 0]
= (sin 0 + cos 0)3,
so that sin 0 +cos = +........................ (5).
Subtracting (4) from (3), we have
x _ = (cos 0 - sin 0) (1- 2 sin 0 cos 0)
a b
=(cos 0 - sin 0)3,
so that cos0 - sin0= --........................ (6).
Squaring and adding (5) and (6), we have
EXAMPLES. XLV.
Eliminate 0 from the equations
1. acos0+bsin =c, and bcosO-asinO=d.
2. x=acos(0-a), and y=bcos(0-3).
3. acos20=b sin 0, and csin20 =dcos0.
4. asina-bcosa=2bsin0, and asin2a-bcos20=a.
5 -.4... - an,sin2 0 cos2 0 1
5. xsin 0-y cos 0=^/x2+y2, and s i + S= 2 -.
b 19-22 2
19 —2