Convert Parametric Equations to a Cartesian Equation #2

[@jan-willemreens9010]

...Miss Jay, I liked converting the same Parametric Equations to a Cartesian Equation in a different way: (Theta=T), (1) x=4+sin(T) ---> sin(T)=x-4, (2) y=1-cos(2T) ---> y=sin^2(T)+cos^2(T)-(cos^2(T)-sin^2(T)) ---> y=2sin^2(T), and thus after substitution of sin(T)=x-4 ---> y=2(x-4)^2. Same result obtained by different trig. identities: 1=sin^2(T)+cos^2(T) and cos(2T)=cos^2(T)-sin^2(T). They look like puzzles for advanced practitioners and I find them sometimes quite tricky... Thank you for this entertaining video, Jan-W

[@MathsWithJay]

There is often more than one way of doing a question when trig identities are involved...

[@jan-willemreens9010]

@@MathsWithJay ...That's why I like experimenting with trigonometry so much... Thank you for your comment, Jan-W

[@jan-willemreens9010]

...Miss Jay, We know that x=4+sin(T) and y=1-cos(2T), and that sin(T) and cos(2T) vary between -1 and 1 where T is a member of all real numbers. Can we not conclude from this that x varies between 3 (4-1) and 5 (4+1), and that y varies between 0 (1-1) and 2 (1--1=1+1), so 3

[@MathsWithJay]

Yes...the restrictions on x and y look good...have you sketched the graph?

[@jan-willemreens9010]

@@MathsWithJay ...Miss Jay, It is a parabola (y=2(x-4)^2) opening upward with domain D=[3,5] and Range R=[0,2]... Jan-W p.s. Starting at (3,2) and ending at (5,2)? Should a direction be indicated in the graph?

[@MathsWithJay]

Yes that't correct....if you were starting with the angle at zero and increasing to 2pi, you could think about a direction...

[@jan-willemreens9010]

@@MathsWithJay ...Miss Jay, Thank you for your constructive comment... Jan-W