Plane trigonometry, by S.L. Loney.

MAXIMUM AND MINIMUM VALUES.

471

We have 2 sin x sin y =2 sin x sin (a - x)
= cos (a - 2x)- cos a.
Hence 2 sin x sin y is greatest when cos (a - 2x) is
greatest, i.e. when a= 2x, and therefore
y
The product is therefore greatest when the angles x
and y are equal.
Let there be three angles x, y, and z whose sum is
equal to a constant angle /. If, in the product
sin x sin y sin z,
any two of the angles x and y be unequal, we can, by the
preceding part of the article, increase the product by
substituting for both x and y half their sum without
increasing or diminishing the sum of the angles.
Hence so long as the angles x, y, and z are unequal,
we can increase the given product by thus making the
angles approach to equality.
The maximum value will therefore be obtained when
the angles x, y, and z are equal.
This argument can clearly be applied whatever be the
number of the angles x, y, z....
400. We can now shew that the maximum triangle
that can be inscribed in a given circle is equilateral.
For, if R be the radius of the circle, we have (as in
Ex. xxxvI. 10) the area of the triangle
= 2R2 sin A sin B sin C,
where A +B + C7= 27r, a constant angle. By the preceding article it follows that the triangle is greatest when
A=B=C.