Pregunta
Consider the following curve.
Write an equation for the curve in terms of
and
Ask by Mcguire Chambers. in the United States
Nov 05,2024
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The equation of the curve in terms of
and
is
.
Solución
To rewrite the equation
in terms of
and
, we can use the double angle identity for cosine:
We also know that in polar coordinates,
, where
and
.
Using the identity for
, we can substitute it into the original equation:
Now, substituting
and using
and
, we can express
and
in terms of
and
:
Thus, we can rewrite
:
Substituting this back into the equation gives:
This simplifies to:
Thus, the equation of the curve in terms of
and
is:
This is the final form of the equation.
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Bonus Knowledge
To rewrite the equation
in terms of
and
, we can use the double angle identity for cosine, which states that
.
Substituting this identity into the original equation, we have:
Now, recalling that in polar coordinates,
(where
and
), we can express
in terms of
and
as:
Finally, to simplify, substitute
with
and
with
, therefore, we find:
This simplifies to give us the relationship involving
and
corresponding to the original polar equation.
In summary, the curve can be represented as:

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