If x=cost and y=tant, how do I get rid of the parameter?

Question

welshfella · Answer

if y is the tangent of t   then t is the angle whose tangent is y
so we can write t = tan-1 y  or t = arctan y

 do a similar procedure with y = cos t   then you can form a relation between x and y

kainui · Answer

I'd do it this way:

$$y=\tan t = \frac{ \sin t}{\cos t}$$

Now we know we can plug in $\cos t =x$ but what about the $\sin t$ part? Well here's a trick, use the Pythagorean identity:

$$\sin^2 t + \cos^2 t = 1$$
We can rearrange to get:

$$\sin t = \sqrt{1- x^2}$$

So now we can finish plugging in:


$$y= \frac{ \sin t}{\cos t} = \frac{\sqrt{1-x^2}}{x}$$

anonymous · Answer

is + or - not necessary here? for the Square Root

welshfella · Answer

I think not because the angles are considered to be only in the first quadrant

kainui · Answer

Oh yeah I think $\pm$ is a good idea @ArkGoLucky

anonymous · Answer

Oh okay. Thanks.

anonymous · Answer

Could you help me with one more problem while you are here?

anonymous · Answer

Well either of you