Find the angle between the radius vector and the tangent for the given curve r²Cos2theta = a²
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This function reaches a maximum at the points
x
=
2
π
n
m
,
n
∈
Z
.
By the periodicity, the curvature at all maximum points is the same, so it is sufficient to consider only the point
x
=
0.
Write the derivatives:
y
′
=
(
cos
m
x
)
′
=
−
m
sin
m
x
,
y
′
′
=
(
−
m
sin
m
x
)
′
=
−
m
2
cos
m
x
.
The curvature of this curve is given by
K
=
|
y
′
′
|
[
1
+
(
y
′
)
2
]
3
2
=
∣
∣
−
m
2
cos
m
x
∣
∣
[
1
+
(
−
m
sin
m
x
)
2
]
3
2
=
∣
∣
−
m
2
cos
m
x
∣
∣
(
1
+
m
2
sin
2
m
x
)
3
2
.
At the maximum point
x
=
0
,
the curvature and radius of curvature, respectively, are equal to
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