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MAA AMC
AMC 12/AHSME
1990 AMC 12/AHSME
21
Pyramid and Trig
Pyramid and Trig
Source:
April 4, 2006
geometry
3D geometry
pyramid
trigonometry
Pythagorean Theorem
Problem Statement
Consider a pyramid
P
−
A
B
C
D
P-ABCD
P
−
A
BC
D
whose base
A
B
C
D
ABCD
A
BC
D
is a square and whose vertex
P
P
P
is equidistant from
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
. If
A
B
=
1
AB=1
A
B
=
1
and
∠
A
P
D
=
2
θ
\angle APD=2\theta
∠
A
P
D
=
2
θ
then the volume of the pyramid is
(A)
sin
θ
6
(B)
cot
θ
6
(C)
1
6
sin
θ
(D)
1
−
sin
2
θ
6
(E)
cos
2
θ
6
sin
θ
\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}
(A)
6
s
i
n
θ
(B)
6
c
o
t
θ
(C)
6
s
i
n
θ
1
(D)
6
1
−
s
i
n
2
θ
(E)
6
s
i
n
θ
c
o
s
2
θ
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