42–43. Intersection points Find the intersection points of the following curves.
r= √(cos3t) and r= √(sin3t)

Verified step by step guidance

1. Rewrite the given curves in terms of their polar coordinates: the first curve is \(r = \sqrt{\cos(3t)}\) and the second curve is \(r = \sqrt{\sin(3t)}\).
2. To find the intersection points, set the two expressions for \(r\) equal to each other: \(\sqrt{\cos(3t)} = \sqrt{\sin(3t)}\).
3. Square both sides to eliminate the square roots, giving \(\cos(3t) = \sin(3t)\).
4. Use the trigonometric identity or rewrite the equation to find values of \(t\) that satisfy \(\cos(3t) = \sin(3t)\), for example by dividing both sides by \(\cos(3t)\) (where defined) to get \$1 = \tan(3t)\(, so \)\tan(3t) = 1$.
5. Solve for \(t\) by finding all angles where \(\tan(3t) = 1\), then substitute these \(t\) values back into either \(r = \sqrt{\cos(3t)}\) or \(r = \sqrt{\sin(3t)}\) to find the corresponding \(r\) values, which together give the intersection points in polar coordinates.

Key Concepts

Polar Coordinates and Parametric Curves

Polar coordinates represent points using a radius and angle (r, t), where r is the distance from the origin and t is the angle. Understanding how curves are defined in polar form, especially with trigonometric functions of multiples of t, is essential to analyze and find intersections.

Solving Equations Involving Trigonometric Functions

Finding intersection points requires solving equations where expressions involving trigonometric functions like cos(3t) and sin(3t) are equal. Knowledge of trigonometric identities and properties helps simplify and solve these equations for t.

Conditions for Intersection in Polar Coordinates

Two polar curves intersect where their radius values are equal for the same angle or where their points coincide in the plane. This involves equating r-values and considering the domain restrictions (e.g., square roots require non-negative arguments) to find valid intersection points.