Trigonometric Elimination

Question

Eliminate θ from the pair of equations:

a) x= acosθ and y= bsinθ 

Answer 1

x = a * cos T

y = b * sin T

squares both of them

x^2 = a^2*cos^2

y^2 = b^2*sin^2

Muljtplies first equation by b^2 and second equation by a^2

b^2*x^2 = a^2*b^2*cos^2

a^2*y^2 = a^2*b^2*sin^2

b^2*x^2 +a^2*y^2 = a^2*b^2*cos^2 + a^2*b^2*sin^2

b^2*x^2 + a^2*y^2 = a^2*b^2 <---- identity: sin^2 + cos^2 = 1

x^2/a^2 + y^2/b^2 = 1

(x/a)^2 + (y/b)^2 = 1

It's an ellipse

Answer 2

I am not sure exactly what is wanted here, but one possibility is

(x/a) = cos theta

(y/b) = sin theta

(x/a)^2 = (cos theta)^2

(y/b)^2 = (sin theta)^2

(x/a)^2 + (y/b)^2 = 1