Find the value of φ for the following curves.

Question image

Understand the Problem

The question is asking to find the value of φ for certain polar curves and to prove a relationship involving these curves.

Answer

The angle \( \phi \) corresponds to the maximum radius of the cardioid at \( \phi = \frac{\pi}{3} \).
Answer for screen readers

The value of ( \phi ) for the curve ( r = 1 + \cos \theta ) involves evaluative properties of the cardioid structure, leading to ( \phi = \frac{\pi}{3} ) at key angles like ( \theta = 0 ).

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

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

  2. 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.

  3. 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: $$ \phi = \tan^{-1}\left(\frac{y}{x}\right) $$

  4. 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 ).

The value of ( \phi ) for the curve ( r = 1 + \cos \theta ) involves evaluative properties of the cardioid structure, leading to ( \phi = \frac{\pi}{3} ) at key angles like ( \theta = 0 ).

More Information

In polar coordinates, cardioids like the one defined can illustrate beautiful symmetry and geometric properties, often associated with waveforms in acoustics and optics. The cardioid shape appears in various natural phenomena and mathematical studies.

Tips

  • Misinterpreting the angle ( \phi ) in terms of ( \theta ); it's crucial to isolate the polar properties correctly.
  • Failing to convert properly from polar coordinates to Cartesian coordinates; ensure both equations are employed correctly.
  • Confusing the maximum point of ( r ) as related to ( \theta ) rather than evaluating the symmetry of the curve.

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