I feel confused.

Plotting t*cos(t) on the x axis and t*sin(t) on the y axis does, indeed, generate a spiral.

There isn't much point in trying to express y in terms of x without the t as there are multiple values (an infinite number) of y for a given x (and vice-versa).

If you plot t*cos(t) on the y axis and t*sin(t) on the x axis you get a spiral that curves in the opposite direction.

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just use the same formula but plug in x for y and y for x

sin^2 + cos^2 = 1

so sin = sqrt (1-cos^2)

y = t cos t        so  y/t =cos t

x = t sin t =  t  ( sqrt (1- y^2/t^2) )      perhaps?   Totally not sure   Good luck !

Well, wait a minute...... that was x in terms of y

x/t =  sin t          y = t (sqrt(1-sin^2 t) = t  sqrt(1-x^2/t^2)          A shot in the dark !!!

 x / t  = cos (t)

[x / t] ^2  = cos^2(t)

[x / t] ^2 = 1 - sin^2(t)

[x / t] ^2  = 1 - [ y/t] ^2

[x / t] ^2 =  [t^2 - y^2] / t^2

x^2  = [t^2 - y^2]

y^2  = t^2 - x^2    take the + root

y  =  √ [ t^2 – x^2]

y  = √ [ t^2 – t^2*cos^2(t) ]

Compare the graphs

Max....maybe you're after somehing like this ???

You might find this of interest :  https://en.wikipedia.org/wiki/Archimedean_spiral

Also.....some polar plots can produce some pretty neat looking graphs :

http://www.mathamazement.com/Lessons/Pre-Calculus/06_Additional-Topics-in-Trigonometry/graphs-of-polar-equations.html

1.

\(y = t\sin t\\ x = t\cos t\)

2.

\(y = t\cos t\\ x = t\sin t\)