Final answer:
To find the highest point on the curve r = sin(2theta), transform to Cartesian coordinates and set the derivative of r to zero to find theta values for maximum r. Subsequently, evaluate y for these theta values to find the exact y-coordinates of the highest points.
Step-by-step explanation:
Finding the Y-Coordinate of the Highest Point on the Curve
To find the y-coordinate of the highest points on the curve r = sin(2theta), we first need to consider the polar to Cartesian coordinate transformation formulas:
y = r sin(theta)
x = r cos(theta)
Since the highest point on the curve corresponds to the maximum r value, we can set the derivative of r with respect to theta equal to zero to find the theta at the peak:
Find dr/dtheta: dr/dtheta = 2cos(2theta).
Set dr/dtheta equal to 0 and solve for theta to find the angles at which r is at a maximum.
Substitute the theta values back into y = r sin(theta) to find the y-coordinates of the highest points.
Note that since the sine function oscillates between -1 and 1, the maximum value of r is 1 when sin(2theta) = 1. Therefore, we only need to find the theta values where this condition is met.
For sin(2theta) = 1, the general solution for theta is theta = (2n+1)pi/4, where n is an integer.
Therefore, we evaluate the y-coordinate using:
y = sin((2n+1)pi/4) when r is at its maximum.
Thus, the exact y-coordinates of the highest points on the curve are given by the sine of these specific angle values.