Graph and identify the kind of conic the graph represents:

a. x(t)=3 cos t
y(t)=3 sin t
on [0,2pi]

b. x(t) =2 cos t
y(t) =5 sin t
on [1,5]

I tried graphing A on my TI-89 calculator as y(1)=3 cos x and y(2)=3 sin x. All I see are two wave-like graphs crossing each other; I don't see a conic. Did I do something wrong?

Thanks very much!

those are parametric equations - x(t) and y(t).

Go to mode, scroll down until you see "Func" selected. Select "Par" instead.

Now you can graph the equations x(t) and y(t) together.

To determine the kind of conic represented by a parametric equation, you need to eliminate the parameter and rewrite the equation in the form of a standard conic equation. In this case, the standard conic equations are:

1. For an ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1
2. For a circle: (x - h)^2 + (y - k)^2 = r^2
3. For a parabola: y = ax^2 + bx + c
4. For a hyperbola: (x - h)^2/a^2 - (y - k)^2/b^2 = 1

Now let's determine the kind of conic for each case:

a. x(t) = 3 cos(t)
y(t) = 3 sin(t)

To eliminate the parameter, we can square both equations:
x^2 = 9 cos^2(t)
y^2 = 9 sin^2(t)

Adding these equations together, we get:
x^2 + y^2 = 9(cos^2(t) + sin^2(t))

Using the trigonometric identity cos^2(t) + sin^2(t) = 1, we simplify further:
x^2 + y^2 = 9

The equation x^2 + y^2 = 9 represents a circle centered at the origin (0,0) with a radius of 3. So, the graph represents a circle.

b. x(t) = 2 cos(t)
y(t) = 5 sin(t)

Again, let's eliminate the parameter by squaring both equations:
x^2 = 4 cos^2(t)
y^2 = 25 sin^2(t)

Adding these equations together, we have:
x^2 + y^2 = 4(cos^2(t) + 25 sin^2(t))

Using the trigonometric identity cos^2(t) + 25 sin^2(t) = 1, we simplify further:
x^2 + y^2 = 4

The equation x^2 + y^2 = 4 represents a circle centered at the origin (0,0) with a radius of 2. So, the graph also represents a circle.

In summary, both graphs represent circles. It seems that you might have made an error while graphing the equations on your calculator. Make sure you input the correct equations and try again.