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Rx5674/沙盒
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<
User:Rx5674
P
=
p
1
+
p
2
+
p
3
=
sin
2
θ
+
sin
2
(
θ
−
2
π
3
)
+
sin
2
(
θ
+
2
π
3
)
=
3
2
{\displaystyle {\begin{aligned}P&=p_{1}+p_{2}+p_{3}\\&=\sin ^{2}\theta +\sin ^{2}\left(\theta -{\frac {2\pi }{3}}\right)+\sin ^{2}\left(\theta +{\frac {2\pi }{3}}\right)\\&={\frac {3}{2}}\end{aligned}}}
P
=
p
1
+
p
2
+
p
3
=
sin
2
θ
+
sin
2
(
θ
−
2
π
3
)
+
sin
2
(
θ
+
2
π
3
)
=
1
2
[
1
−
cos
2
θ
+
1
−
cos
2
(
θ
−
2
π
3
)
+
1
−
cos
2
(
θ
+
2
π
3
)
]
=
1
2
[
3
−
cos
2
θ
−
cos
2
(
θ
−
2
π
3
)
−
cos
2
(
θ
+
2
π
3
)
]
{\displaystyle {\begin{aligned}P&=p_{1}+p_{2}+p_{3}\\&=\sin ^{2}\theta +\sin ^{2}\left(\theta -{\frac {2\pi }{3}}\right)+\sin ^{2}\left(\theta +{\frac {2\pi }{3}}\right)\\&={\frac {1}{2}}[1-\cos 2\theta +1-\cos 2(\theta -{\frac {2\pi }{3}})+1-\cos 2(\theta +{\frac {2\pi }{3}})]\\&={\frac {1}{2}}[3-\cos 2\theta -\cos 2(\theta -{\frac {2\pi }{3}})-\cos 2(\theta +{\frac {2\pi }{3}})]\\\end{aligned}}}
P
=
p
1
+
p
2
=
sin
2
θ
+
sin
2
(
θ
+
π
2
)
=
1
2
[
1
−
cos
2
θ
+
1
−
cos
2
(
θ
+
π
2
)
]
=
1
2
[
2
−
2
cos
(
3
θ
+
π
2
)
cos
(
θ
−
π
2
)
]
{\displaystyle {\begin{aligned}P&=p_{1}+p_{2}\\&=\sin ^{2}\theta +\sin ^{2}(\theta +{\frac {\pi }{2}})\\&={\frac {1}{2}}[1-\cos 2\theta +1-\cos 2(\theta +{\frac {\pi }{2}})]\\&={\frac {1}{2}}[2-2\cos(3\theta \ +{\frac {\pi }{2}})\cos(\theta -{\frac {\pi }{2}})]\\\end{aligned}}}
P
=
p
1
+
p
2
=
sin
2
θ
+
sin
2
(
θ
+
π
2
)
=
sin
2
θ
+
cos
2
θ
=
1
{\displaystyle {\begin{aligned}P&=p_{1}+p_{2}\\&=\sin ^{2}\theta +\sin ^{2}(\theta +{\frac {\pi }{2}})\\&=\sin ^{2}\theta +\cos ^{2}\theta \\&=1\end{aligned}}}