Programming Team Independent Study Homework Week #7: Trigonometry
1) In triangle ABC, show that sin(A/2) ≤ a/(b+c).
2) A circle of radius 1 is randomly placed in a 15-by-36 rectangle ABCD so that the circle
lies completely within the rectangle. Given that the probability that the circle will not
touch diagonal AC is m/n, where m and n are relatively prime positive integers, find
m+n.
3) Prove that tan 3a – tan 2a – tan a = tan3atan2atana, for all a that are not integer multiples
of π/2.
4) In triangle ABC, show that
a.
b.
c.
d.
e.
4R = abc/[ABC].
2R2sinAsinBsinC = [ABC]
2RsinAsinBsinC = r(sinA+sinB+sinC)
R = 4Rsin(A/2)sin(B/2)sin(C/2)
acosA+bcosB+ccosC = abc/(2R2).
Note: R = circumradius, r = inradius, [ABC] = area triangle ABC.
5) Given that
, determine n.
6) Let ABC be a triangle such that
where s and r denote its semiperimeter and its inradius, respectively. Prove that ABC is
similar to a triangle T whose side lengths are all positive integers with no common
divisor and determine these integers.
7) Triangle ABC has the following property: there is an interior point P such that angle PAB
= 10 degrees, angle PBA = 20 degrees, angle PCA = 30 degrees and angle PAC = 40
degrees. Prove that triangle ABC is isosceles.