To analyze the shape of the graph represented by the polar equation

$3r \sin(\theta) + 5r^2 \cos^2(\theta) = 2$

we can try rewriting it in a more familiar form or investigating its key properties.

Step 1: Simplify the Equation

Rewrite the equation in terms of $r$ and trigonometric identities: $3r \sin(\theta) + 5r^2 \cos^2(\theta) = 2.$ Notice that the second term, $5r^2 \cos^2(\theta)$, could be more manageable if we change it to $r^2 \cos^2(\theta) = (r \cos(\theta))^2$. This substitution helps us because $x = r \cos(\theta)$ and $y = r \sin(\theta)$ in Cartesian coordinates.

Step 2: Convert to Cartesian Coordinates

Using $x = r \cos(\theta)$ and $y = r \sin(\theta)$, the terms translate as follows:

• $r \sin(\theta) = y$
• $r \cos(\theta) = x$, thus $r^2 \cos^2(\theta) = x^2$.

Substitute these into the equation: $3y + 5x^2 = 2.$

Step 3: Rearrange the Equation

Now, we have the equation in Cartesian form: $5x^2 + 3y = 2.$ Solving for $y$ gives: $y = \frac{2 - 5x^2}{3}.$

Step 4: Analyze the Shape

This equation $y = \frac{2 - 5x^2}{3}$ resembles the form of a parabola. The term $-5x^2$ indicates that it opens downward along the $y$-axis. Hence, the graph will be a parabola opening downward with a vertex at $\left(0, \frac{2}{3}\right)$.

Conclusion

The graph of the given polar equation represents a downward-opening parabola.

Would you like further details on the derivation, or have any questions?

---

Here are some related questions for further exploration:

1. How can we determine if a polar equation represents a conic section?
2. What effect do coefficients have on the shape of a polar graph?
3. How would the equation change if we replaced $\cos^2(\theta)$ with $\sin^2(\theta)$?
4. How can we convert between polar and Cartesian coordinates systematically?
5. What other methods can be used to analyze the shape of polar equations?

Tip: Converting polar equations to Cartesian form can simplify analysis by transforming unfamiliar terms into more recognizable ones.