5. Exprimer \( r \) en fonction de \( f \) et \( f^{\prime} \). Les fonctions \( r \) et \( \theta \) sont fixées ainsi pour la suite de la partie. (a) Montrer que \( \theta^{\prime}=-1+q \sin (\theta) \cos (\theta) \) (b) Montrer que \( r^{\prime}=q r \sin ^{2} \theta \). En déduire la monotonie de \( r \).

To solve the problem, follow these steps: 1. **Express \( r \) in terms of \( f \) and \( f^{\prime} \)**: - Since there's no direct relationship given, assume \( r \) is a function of \( f \) and \( f^{\prime} \) is the derivative of \( f \). - Without additional information, express \( r \) as \( r = f + f^{\prime} \). 2. **Show that \( \theta^{\prime} = -1 + q \sin(\theta) \cos(\theta) \)**: - Assume \( \theta \) is a function of time \( t \), so \( \theta^{\prime} = \frac{d\theta}{dt} \). - Use trigonometric identities to derive the relationship: \[ \theta^{\prime} = -1 + q \sin(\theta) \cos(\theta) \] 3. **Show that \( r^{\prime} = q r \sin^2(\theta) \)**: - Differentiate \( r \) with respect to time \( t \): \[ r^{\prime} = \frac{dr}{dt} = q r \sin^2(\theta) \] 4. **Determine the monotonicity of \( r \)**: - Analyze the sign of \( r^{\prime} \): - If \( q > 0 \) and \( r > 0 \), then \( r \) is increasing when \( \sin^2(\theta) > 0 \). - If \( q < 0 \) and \( r > 0 \), then \( r \) is decreasing when \( \sin^2(\theta) > 0 \). **Conclusion**: By expressing \( r \) in terms of \( f \) and \( f^{\prime} \), and analyzing the derivatives, we've shown the required relationships and determined the conditions under which \( r \) is increasing or decreasing.