103 Trigonometry Problems

206 103 Trigonometry Problems
The inequality is a special case of the power mean inequality.
Schur’s Inequality
Let x,y,z be nonnegative real numbers. Then for any r>0,
x r (x − y)(x − z) + y r (y − z)(y − x) + z r (z − x)(z − y) ≥ 0.
Equality holds if and only if x = y = z or if two of x,y,z are equal and the third is
equal to 0.
The proof of the inequality is rather simple. Because the inequality is symmetric
in the three variables, we may assume without loss of generality that x ≥ y ≥ z.
Then the given inequality may be rewritten as
(x − y) [ x r (x − z) − y r (y − z) ] + z r (x − z)(y − z) ≥ 0,
and every term on the left-hand side is clearly nonnegative. The first term is positive
if x>y, so equality requires x = y, as well as z r (x − z)(y − z) = 0, which gives
either x = y = z or z = 0.
Sector
The region enclosed by a circle and two radii of the circle.
Stewart’s Theorem
In a triangle ABC with cevian AD, write a =|BC|, b =|CA|, c =|AB|, m =
|BD|, n =|DC|, and d =|AD|. Then
d 2 a + man = c 2 n + b 2 m.
This formula can be used to express the lengths of the altitudes and angle bisectors
of a triangle in terms of its side lengths.
Trigonometric Identities
sin 2 a + cos 2 a = 1,
1 + cot 2 a = csc 2 a,
tan 2 x + 1 = sec 2 x.