9. Transform each polar equation to an equation in rectangular coordinates and identify its shape. ( 4 points) a. \( \theta=1.34 \) radians b. \( r=\tan \theta \sec \theta \)

For \( \theta = 1.34 \) radians, we can convert this polar equation to rectangular coordinates using the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). In this case, \( r \) can be expressed as \( r = \frac{y}{\sin 1.34} = \frac{x}{\cos 1.34} \). This results in the equation representing a straight line emanating from the origin, specifically at an angle of 1.34 radians from the positive x-axis. For \( r = \tan \theta \sec \theta \), we can rewrite the equation using the definitions of tangent and secant. We know that \( \tan \theta = \frac{y}{x} \) and \( \sec \theta = \frac{1}{\cos \theta} = \frac{r}{x} \). Substituting these gives us \( r = \frac{y \cdot r}{x^2} \). Simplifying this leads to the rectangular coordinate equation \( x^2 = y \cdot r \), revealing a curve that represents a parabola opening to the right.