Equations written in polar form can be converted into rectangular (Cartesian) form using the relationships between polar and rectangular coordinates. In what follows, the polar coordinates of a point are denoted by \( (R, t) \), where \( R \) is the radial distance and \( t \) is the angular coordinate.
The relationships between rectangular coordinates \( (x, y) \) and polar coordinates \( (R, t) \) are:
\[ R^2 = x^2 + y^2, \qquad x = R \cos t, \qquad y = R \sin t \]Convert the polar equation
\[ R = 4 \sin t \]to rectangular form.
Multiply both sides of the equation by \( R \):
\[ R^2 = 4R \sin t \]Using the identities \( R^2 = x^2 + y^2 \) and \( y = R \sin t \), we rewrite the equation as:
\[ x^2 + y^2 = 4y \]Rearranging:
\[ x^2 + y^2 - 4y = 0 \]This is the equation of a circle.
Convert the polar equation
\[ R(-2 \sin t + 3 \cos t) = 2 \]to rectangular form.
First, expand the left-hand side:
\[ -2R \sin t + 3R \cos t = 2 \]Using the substitutions \( y = R \sin t \) and \( x = R \cos t \), the equation becomes:
\[ -2y + 3x = 2 \]This is the equation of a straight line.
Convert the polar equation
\[ t + \frac{\pi}{4} = 0 \]to rectangular form.
Solve the equation for \( t \):
\[ t = -\frac{\pi}{4} \]Take the tangent of both sides:
\[ \tan t = \tan\left(-\frac{\pi}{4}\right) = -1 \]Using the identity \( \tan t = \dfrac{\sin t}{\cos t} \) and the relationships \( x = R \cos t \), \( y = R \sin t \), we obtain:
\[ \tan t = \frac{y}{x} \]Therefore:
\[ \frac{y}{x} = -1 \]or equivalently:
\[ y = -x \]This is the equation of a straight line.