Final answer:
The solution to the system of equations r = cos(3θ) and r = sin(3θ) occurs where cos(3θ) equals sin(3θ), which is at θ = 15° and θ = 75°.
Step-by-step explanation:
The student is asking to find the solution to the system of polar equations r = cos(3θ) and r = sin(3θ). To solve this system, one would typically look for values of θ for which both equations produce the same value for r. In polar coordinates, the value of r represents the distance from the origin to a point, while θ represents the angle that line makes with the x-axis.
For both equations to be true, cos(3θ) must equal sin(3θ). This equality holds when 3θ is an odd multiple of 45° (or π/4 radians), because sine and cosine functions are co-terminal at these angles. Solving for θ gives us the solutions to the system. The main angles where sine and cosine are equal are at 45° and 225° (or π/4 and 5π/4 radians), and since we are dealing with 3θ, we divide these by 3 to find the solutions for θ.
Thus, the solutions for θ in the original equation are 15° (or π/12 radians) and 75° (or 5π/12 radians). This result shows the angles at which the length from the origin (r) and the direction (θ) are the same for the both the sine and cosine components of the system's equations.