Answer:
[tex]\dfrac{y^2}{9}=1-\dfrac{x^2}{25}[/tex]
Step-by-step explanation:
When converting parametric equations that involve trig functions to Cartesian equations, use trig identities to eliminate the parameter.
Given parametric equations:
[tex]x=5 \sin \theta, \quad y=3 \cos \theta, \quad 0\leq \theta\leq \pi[/tex]
Square the equation for x:
[tex]\implies x^2=(5 \sin \theta)^2=25 \sin^2 \theta[/tex]
Use the identity [tex]\sin^2 x+\cos^2x =1[/tex] to write [tex]x^2[/tex] in terms of cos:
[tex]\implies x^2=25(1-\cos^2 \theta)[/tex]
Isolate [tex]\cos^2 \theta[/tex] :
[tex]\implies \dfrac{x^2}{25}=1-\cos^2 \theta[/tex]
[tex]\implies \cos^2 \theta=1- \dfrac{x^2}{25}[/tex]
Square the equation for y:
[tex]\implies y^2=(3 \cos \theta)^2=9 \cos ^2 \theta[/tex]
Replace [tex]\cos^2 \theta[/tex] with the found equation involving [tex]x^2[/tex] :
[tex]\implies y^2=9\left(1-\dfrac{x^2}{25}\right)[/tex]
Divide both sides by 9:
[tex]\implies \dfrac{y^2}{9}=1-\dfrac{x^2}{25}, \quad \textsf{with }x\geq 0[/tex]
