11. graph x = cos (2θ), y = 2 sin θ using -90° ≤ θ ≤ 90°.

11. graph x = cos (2θ), y = 2 sin θ using -90° ≤ θ ≤ 90°.

11. graph x = cos (2θ), y = 2 sin θ using -90° ≤ θ ≤ 90°.

Answer

Answer:

To graph the parametric - equations (x = \cos(2\theta)) and (y = 2\sin\theta) for (-90^{\circ}\leq\theta\leq90^{\circ}), we can follow these steps:

  1. Recall the double - angle formula (\cos(2\theta)=1 - 2\sin^{2}\theta). Since (y = 2\sin\theta), then (\sin\theta=\frac{y}{2}). Substituting (\sin\theta) into the equation for (x), we get (x = 1-2(\frac{y}{2})^{2}=1-\frac{y^{2}}{2}).

  2. Now, find some key points by choosing values of (\theta):

    • When (\theta=-90^{\circ}), (x=\cos(- 180^{\circ})=-1) and (y = 2\sin(-90^{\circ})=-2).
    • When (\theta=-45^{\circ}), (x=\cos(-90^{\circ}) = 0) and (y=2\sin(-45^{\circ})=-\sqrt{2}\approx - 1.414).
    • When (\theta = 0^{\circ}), (x=\cos(0^{\circ}) = 1) and (y = 2\sin(0^{\circ})=0).
    • When (\theta = 45^{\circ}), (x=\cos(90^{\circ}) = 0) and (y=2\sin(45^{\circ})=\sqrt{2}\approx1.414).
    • When (\theta = 90^{\circ}), (x=\cos(180^{\circ})=-1) and (y = 2\sin(90^{\circ})=2).

  3. Plot these points ((-1,-2),(0,-\sqrt{2}),(1,0),(0,\sqrt{2}),(-1,2)) on the coordinate plane and then connect them with a smooth curve. The resulting graph is a parabola opening to the left with the vertex at ((1,0)).

Explanation:

Step1: Use double - angle formula

We use (\cos(2\theta)=1 - 2\sin^{2}\theta) to get a non - parametric form.

Step2: Substitute (\sin\theta)

Substitute (\sin\theta=\frac{y}{2}) into the (x) equation: (x = 1-\frac{y^{2}}{2}).

Step3: Find key points

Evaluate (x) and (y) for (\theta=-90^{\circ},-45^{\circ},0^{\circ},45^{\circ},90^{\circ}).

Step4: Plot and connect

Plot the points and draw a smooth curve.