Plane trigonometry, by S.L. Loney.

ELIMINATION.

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246. Again, suppose we knew that an angle 0
satisfied both of the equations
sin O = b, and cos3 0 = c,
so that sin = b3, and cos 0 = c.
Now we always have, for all values of 0,
sin2 0 + cos2 = 1,
so that in this case b3 + c3 = 1.
This is the result of eliminating 0.
247. Between any two equations involving one
unknown quantity we can, in theory, always eliminate
that quantity. In practice a considerable amount of
artifice and ingenuity is often required in seemingly
simple cases.
So between any three equations involving two unknown quantities we can theoretically eliminate both
of the unknown quantities.
248. Some examples of elimination are appended.
Ex. 1. Eliminate 0 from the equations
a cos + bsin O=c,
and b cos +c sin 0 =a.
Solving for cos0 and sin 0 by cross multiplication, or otherwise,
we have
cos 0 sin 0 1
2- ab - a-bc - ac -b2
( c2 - ab)2 + (a2 - bc)2
i=e. I -os2 + Si 2(-2 -,
(ac-b2)2
i.e. (a2 - be)2 + (c2 - ab)2= (b2 - ac)2.

L. T.

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