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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1964 Vietnam National Olympiad
1
cos a + cos (a +2\pi /3 ) + cos (a+4\pi /3)
cos a + cos (a +2\pi /3 ) + cos (a+4\pi /3)
Source: Vietnamese MO (VMO) 1964
August 22, 2018
trigonometry
Sum
geometry
Problem Statement
Given an arbitrary angle
α
\alpha
α
, compute
c
o
s
α
+
c
o
s
(
α
+
2
π
3
)
+
c
o
s
(
α
+
4
π
3
)
cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)
cos
α
+
cos
(
α
+
3
2
π
)
+
cos
(
α
+
3
4
π
)
and
s
i
n
α
+
s
i
n
(
α
+
2
π
3
)
+
s
i
n
(
α
+
4
π
3
)
sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)
s
in
α
+
s
in
(
α
+
3
2
π
)
+
s
in
(
α
+
3
4
π
)
. Generalize this result and justify your answer.
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