Converting between Trig Forms to Represent a Cardioid

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SUMMARY

The cardioid defined by the polar equation r = a(1 + cos(θ)) can be equivalently expressed as r = 2a cos²(θ/2) for the interval 0 ≤ θ ≤ 2π. This transformation utilizes trigonometric identities to facilitate the conversion between the two forms. The discussion highlights the importance of understanding polar to Cartesian conversions and the application of trigonometric identities in simplifying expressions. Participants emphasized the need for clarity in equating the two forms to confirm their equivalence.

PREREQUISITES
  • Understanding of polar coordinates and equations
  • Familiarity with trigonometric identities
  • Knowledge of Cartesian coordinate conversions
  • Basic calculus concepts related to curves
NEXT STEPS
  • Study trigonometric identities relevant to polar equations
  • Learn about converting polar equations to Cartesian form
  • Explore the properties of cardioids and their graphical representations
  • Investigate advanced polar coordinate transformations
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in polar coordinates and their applications in geometry.

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Homework Statement



Show that the cardioid r=a(1+cos(theta)) can be represented by r=2acos2(theta/2), 0<=theta<=2pi (theta is between 0 and 2pi).

Homework Equations





The Attempt at a Solution



I'm pretty sure I have to equate the two expressions, but I haven't been able to do this. Then I thought of converting both of them to cartesian form (and seeing if they have the same equation) but that's turning into a bit of a mess. Any guidance would be greatly appreciated!
 
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Those are trig functions. There are lots of identities involving trig functions. One of them might be helpful.
 

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