Problem 90 - Algebra and Trigonometry

Question

Eliminate the parameter: \(x=\cos ^{3} t\) and \(y=\sin ^{3} t\)

Step-by-Step Solution

Answer

The equation after eliminating the parameter 't' is \(x^2 + y^2 - 3xy = 0\).

Step 1: Express sin t and cos t in terms of x and y

It is given that \(x = \cos^{3}t\) and \(y = \sin^{3}t\). We can express \(\cos t\) and \(\sin t\) in terms of x and y by taking the cube root of both sides of the given equations. So, we have \(\cos t = x^{1/3}\) and \(\sin t = y^{1/3}\).

Step 2: Use the Pythagorean identity

Now we substitute \(\cos t = x^{1/3}\) and \(\sin t = y^{1/3}\) into the Pythagorean identity \(\sin^{2}t + \cos^{2}t = 1\) to obtain the equation \((x^{1/3})^{2} + (y^{1/3})^{2} = 1\). Simplifying this gives \(x^{2/3} + y^{2/3} = 1\).

Step 3: Final simplification

The equation as it stands is correct but quite complex and can be made simpler. We can simplify the equation further by cubing both sides of the equation. This gives the final answer as \(x^2 + y^2 - 3xy = 0\).