Write the pair of parametric equations in rectangular form:

Question

x=2tant/2 ; y=sect/2

Answer 1

x = 2 tan(t/2) and y = sec(t/2) so y = 1/[ cos(t/2)] or cos(t/2)= 1/y

cos^-1(1/y) = t/2 so t = 2 cos^-1(1/y)

x = 2 tan (t/2) so x = 2 tan[ (2cos^-1(1/y)) / 2 ] = 2 tan[ cos^-1(1/y) ] = 2 tan(θ)

If θ = cos^-1(1/y), then cos(θ) = 1/y and this triangle represents that situation

and so the tan(θ) = sqrt(y²-1)

And now we have

x = 2*sqrt(y²-1)

And that is an equation in rectangular form, it can be simplified to a "y=" equation...

x² = 4(y²-1)

x² / 4 = y² -1

x² / 4 + 1 = y²

Or y = ± sqrt( x² /4 + 1)

Answer 2

To convert the parametric equations to rectangular form requires doing the following. Let x/2 = tan(t/2) and

y = sec(t/2) now use the trig identity 1 + tan²θ = sec²θ here θ = t/2. So 1 + x²/4 = y². So 4y² -x² = 4 is the conversion.