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MAA AMC
AMC 12/AHSME
2021 AMC 12/AHSME Fall
10
10
Part of
2021 AMC 12/AHSME Fall
Problems
(1)
MAA be simping for isosceles triangles this year
Source: 12B #10
11/17/2021
What is the sum of all possible values of
t
t
t
between
0
0
0
and
360
360
360
such that the triangle in the coordinate plane whose vertices are
(
cos
4
0
∘
,
sin
4
0
∘
)
,
(
cos
6
0
∘
,
sin
6
0
∘
)
,
(\cos 40 ^{\circ}, \sin 40 ^{\circ}), (\cos 60 ^{\circ}, \sin 60 ^{\circ}),
(
cos
4
0
∘
,
sin
4
0
∘
)
,
(
cos
6
0
∘
,
sin
6
0
∘
)
,
and
(
cos
t
∘
,
sin
t
∘
)
(\cos t ^{\circ}, \sin t ^{\circ})
(
cos
t
∘
,
sin
t
∘
)
is isosceles?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
100
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
150
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
330
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
360
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
380
<span class='latex-bold'>(A)</span>\ 100 \qquad<span class='latex-bold'>(B)</span>\ 150 \qquad<span class='latex-bold'>(C)</span>\ 330 \qquad<span class='latex-bold'>(D)</span>\ 360 \qquad<span class='latex-bold'>(E)</span>\ 380
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
150
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
330
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
360
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
380
AMC
AMC 12
AMC 12 B