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Mathematics
Mathematics
Problem
Consider the five angles
θ
k
=
π
5
+
2
k
π
5
,
k
=
0
,
1
,
2
,
3
,
4.
\theta_k=\frac{\pi}{5}+\frac{2k\pi}{5},\quad k=0,1,2,3,4.
θ
k
=
5
π
+
5
2
k
π
,
k
=
0
,
1
,
2
,
3
,
4.
(a) Express each
θ
k
\theta_k
θ
k
as an equivalent angle lying in the interval
[
0
,
2
π
)
[0,2\pi)
[
0
,
2
π
)
. (b) Hence, or otherwise, show that
sin
θ
0
+
sin
θ
1
+
sin
θ
2
+
sin
θ
3
+
sin
θ
4
=
0.
\sin\theta_0+\sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4=0.
sin
θ
0
+
sin
θ
1
+
sin
θ
2
+
sin
θ
3
+
sin
θ
4
=
0.
Give full reasoning in your proof.
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