Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = \cos \theta$ $y = 5 \sin 2 \theta$
A) $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 5 } = 1$
B) $y = \pm 10 x \sqrt { 1 + x ^ { 2 } }$
C) $y = \pm 10 x \sqrt { 1 + x }$
D)$\frac { x ^ { 2 } } { 1 } - \frac { y ^ { 2 } } { 5 } = 1$
E) $y = \pm 10 x \sqrt { 1 - x ^ { 2 } }$

Question

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.  $x = \cos \theta$ $y = 5 \sin 2 \theta$ 
A) $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 5 } = 1$
B) $y = \pm 10 x \sqrt { 1 + x ^ { 2 } }$
C) $y = \pm 10 x \sqrt { 1 + x }$
D)$\frac { x ^ { 2 } } { 1 } - \frac { y ^ { 2 } } { 5 } = 1$
E) $y = \pm 10 x \sqrt { 1 - x ^ { 2 } }$

Quizplus · Accepted Answer

To eliminate the parameter $\theta$, we use the identities $x = \cos \theta$ and $y = 5 \sin 2 \theta$. Since $\sin 2 \theta = 2 \sin \theta \cos \theta$, we can express $y$ in terms of $x$ by substituting $\cos \theta = x$ and using the double angle formula. This gives $y = 10x\sqrt{1 - x^2}$, where $\sqrt{1 - x^2}$ comes from $\sin^2 \theta + \cos^2 \theta = 1$, solving for $\sin \theta$.

Eliminate the Parameter and Write the Corresponding Rectangular Equation Whose x=cos⁡θx = \cos \thetax=cosθ

Eliminate the Parameter and Write the Corresponding Rectangular Equation Whose $x = \cos \theta$