103 Trigonometry Problems

136 103 Trigonometry Problems
Solution: By the double-angle formulas, the above inequalities reduce to
π
> 2 sin x(cos x − cos y) + 2 sin y(cos y − cos z) + 2 sin z cos z,
2
or
π
> sin x(cos x − cos y) + sin y(cos y − cos z) + sin z cos z.
4
As shown in Figure 5.1, in the rectangular coordinate plane, we consider points
O = (0, 0), A = (cos x,sin x), A 1 = (cos x,0), B = (cos y,sin y), B 1 =
(cos y,0), B 2 = (cos y,sin x), C = (cos z, sin z), C 1 = (cos z, 0), C 2 =
(cos z, sin y), and D = (0, sin z). Points A, B, and C are in the first quadrant
of the coordinate plane, and they lie on the unit circle in counterclockwise
order.
D
C
B
C2
B2
A
O
C1
B1
A1
Figure 5.1.
Let D denote the region enclosed by the unit circle in the first quadrant (including
the boundary). It is not difficult to see that quadrilaterals AA 1 B 1 B 2 ,
BB 1 C 1 C 2 , and CC 1 OD are nonoverlapping rectangles inside region D. Itis
also not difficult to see that [D] = π 4 , [AA 1B 1 B 2 ]=sin x(cos x − cos y),
[BB 1 C 1 C 2 ]=sin y(cos y − cos z), and [CC 1 OD]=sin z cos z, from which
our desired result follows.
15. For a triangle XYZ, let r XYZ denote its inradius. Given that the convex pentagon
ABCDE is inscribed in a circle, prove that if r ABC = r AED and
r ABD = r AEC , then triangles ABC and AED are congruent.
Solution: Let R be the radius of the circle in which ABCDE is inscribed.