Conver the Equation $r=a \sin x+b\cos x$ into Cartesian Form

Hello :)) Wanted to convert the equation $r=a \sin x+b\cos x$, where $a, b \in \mathbb{R}$. Knowing that $x = r \cos x$ and $y= = r \sin x$, I can make $x = (a \sin x + b \cos x) \cos x$. But this is kinda a dead end because I can't simplify this further. Also, knowing that $r = \sqrt{x^2 + y ^2}$, I can make $\sqrt{x^2 + y^2}= a \sin x + b \cos x$. This also seems like a dead end...so, any suggestions?

Solution 1:

I believe you meant to write $\sin\theta$ and $\cos\theta$. To have an equation in Cartesian form, you just want all of the variables to be in terms of $x$ and $y$.

So,

\begin{align*} r=a\sin\theta+b\cos\theta&\implies r^2=ar\sin\theta+br\cos\theta&\text{multiplying both sides by }r\\ &\implies x^2+y^2=ay+bx \end{align*}

From here, you can try to simplify write it in a form you would like, but as it stands, the last line is in Cartesian form.