Find the value of φ for the following curves.

Steps to Solve

1. Identify the polar curve The given polar curve is ( r = 1 + \cos \theta ). This is known as a cardioid.

2. Convert polar to Cartesian coordinates To analyze the relationship, convert from polar to Cartesian coordinates. Recall that: $$ x = r \cos \theta $$ $$ y = r \sin \theta $$

Substituting ( r ): $$ x = (1 + \cos \theta) \cos \theta $$ $$ y = (1 + \cos \theta) \sin \theta $$

3. Express ( x ) and ( y ) in terms of ( r ) We can express ( r ) in terms of ( x ) and ( y ): $$ r = \sqrt{x^2 + y^2} $$

4. Explore the properties of the cardioid Investigate the key properties of the cardioid to find ( \phi ). Notably, for a cardioid ( r = a(1 + \cos \theta) ), when ( \theta = 0 ), ( r ) reaches its maximum.

5. Calculate the angle φ For cardioids, ( \phi ) (the angle corresponding to the horizontal distance) can be derived from its geometry. The maximum ( r ) occurs at ( \theta = 0 ), thus: $$ r_{\text{max}} = 2 $$ Therefore, we evaluate: $$ \phi = \tan^{-1}\left(\frac{y}{x}\right) $$

6. Proving the relationship Use the derived values to show the relationship: $$ dr = \frac{d(r)}{d\phi} = \frac{2a \sin \phi}{d\phi} $$ Confirming the expression that connects ( r ) and ( \phi ).