How to find the $ y$ value when $ x=1$ on $r=2\theta+2cos\theta$

Question

Question

Consider the polar graph $r=2\theta+2\cos\theta$ for $0<\theta<\pi$. On that interval, there is exactly one point on the curve with an $x$-coordinate equal to $1$. Give the $y$-coordinate of that point accurate to three decimal places.

So I plug this into a calculator and try using the trace function and realize that It is impossible to use the trace function, I was wondering if there was any way on the calculator to do this?

Answer 1

I don't know the trace function but, from $x=r \cos \theta$ we have that $x=1$ when: $$ (2\theta+2\cos \theta) \cos \theta=1 $$

WolframAlpha finds a solution of this equation in $0<\theta< \pi$ that is: $\theta=1.2456270775...$. Now use this value to find $y=(2\theta+2\cos \theta) \sin \theta$

Answer 2

From $1 = x = r \cos \theta = (2 \theta + 2 \cos\theta) \cos\theta $ find numerically

$$ \begin{cases} \theta & = -4.60608105432367 \\ \theta & = -1.81609268044352 \\ \theta & = 1.24562707756920 \\ \theta & = 4.81427421305973 \end{cases} $$

Then $y = (2\theta +2 \cos \theta) \sin \theta $