Fairly simple trig question

Question

Three points at coordinates $(0,c)$, $(p,q)$, $(0,d)$ respectively. The angle at $(p,q)$ between $(0,c)$ and $(0,d)$ is $θ$. Find $d$.

_{P.s. This isn't homework.}

Answer 1

Hint :

Let $A(0,c)$, $B(0,d)$, and $C(p,q)$. Length $AB=c-d$, \begin{align} AC^2=p^2+(c-q)^2 \end{align} and \begin{align} BC^2=p^2+(q-d)^2 \end{align} Using cosine formula, we get \begin{align} AB^2&=AC^2+BC^2-2\cdot AC\cdot AB\cos\theta\\ (c-d)^2&=p^2+(c-q)^2+p^2+(q-d)^2-2\sqrt{(p^2+(q-d)^2)(p^2+(q-d)^2)}\cos\theta\\ \end{align}