Show that the curvex = 2 cos t, y = 3 sin t cos t has two tangents at (0, 0) and find their equations.

This is the case of parametric derivatives in which x and y are expressed as functions of a variable t.

The first derivative implies that:



So we need to find both derivatives as functions of the variable t, then:






Thus:



At (0,0)  and , so:


∴

There are two values between -π and π which satisfy these equations simultaneously, namely:
 π/2
-π/2

The equation of a straight line given a point and its slope is


Given that the point is (0,0), then the equation can be written as:


So we will find two straight lines:


For :



For :



Lastly:

Multiplication property of zero