We can use the idea of polar curves So instead of x y we will have r r will represent the length and the angle Consider r f cos θ x r 1 sin θ y r when r 2 x2 y 2 where x y and r are real numbers 2 θ cos 1 x r And sin 1 y r Also tan 1 y x On the other hand z x iy rxr iryr From 1 and 2 This becomes z r cos i sin It is presupposed knowledge that z r cos i sin rei we can find the intersections of different curves more easily and identify the nature of the curve in a more efficient way rather than directly equate them Continuing with the previous result z rei if this equation represents the points then we can find the point of intersection by equating two such different z Here we have to remember that this is for the function r f If r f and t k k is a function then to find the common intersection points we can equate the two points represented by z reior z teiwhere and represent angles Then rei tei In such a scenario both can be equal only if r t and 2n n ℤ Or if r t and when 2n n ℤ If put differently f ei k 2n ei 2n or f ei k 2n ei 2n f k 2n or f k 2n because ei2n 1and ein 1 Polar curves are usually used for complex numbers because of its ease of use in case of complex algebra but here complex numbers are used to represent a point even though i is not directly involved in this problem The number of points of intersection of the two rose curves given by sin p rand cos q r can be found out by equating them as shown previously This is only for intersection if both p and q are odd positive integers The results can be extrapolated to other cases too by changing the bounds below cos p sin q sin q cos 2 q cos p cos 2 q

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