Posts Tagged trigonometry

Building a Picnic Table

Someone writes

Jonathan and/or [redacted],

I’ve got a basic geometry question for either of you, that I need a little help with…

I am working on making a picnic table with [redacted] and we wanted to design our own legs for the table which are angled and cross. I have been able to work out all relationships between heights, lengths, widths, angles. etc. and now have two equations with two unknowns that will allow me to solve for the correct angle at which to cut the wood for the legs. Unfortunately I have forgotten how to solve basic cos, sin, and tan when variables are involved. The two equations are (all units are in inches, not that it matters):

$\displaystyle \textrm{sin} (\alpha)$ $\textstyle =$ $\displaystyle \frac{5.5}{x}$ (1)
$\displaystyle \textrm{tan} (\alpha)$ $\textstyle =$ $\displaystyle \frac{29}{35.5-x}$ (2)

How do I solve for $x$ and $\alpha$? Also, what is the answer 🙂

Thanks,
[redacted]

Because

\begin{eqnarray*}\textrm{tan} (\alpha) & = & \frac{\textrm{sin} (\alpha)}{\text......rac{\textrm{sin} (\alpha)}{\sqrt{1 - \textrm{sin}^{2} (\alpha)}}\end{eqnarray*}

we have

$\displaystyle \frac{29}{35.5-x}$ $\textstyle =$ $\displaystyle \frac{\frac{5.5}{x}}{\sqrt{1 - \left(\frac{5.5}{x}\right)^{2}}}$ (3)

Simplifying and solving by computer algebra, we get

\begin{eqnarray*}\frac{3243}{841} x^2 + \frac{8591}{841} x - \frac{1017005}{3364} & = & 0\end{eqnarray*}

and

\begin{eqnarray*}x & = & \frac{ \pm 1276\sqrt{2071} - 8591}{6486}\end{eqnarray*}

Only the positive choice is a root of Eqns. (1) and (2), giving

\begin{eqnarray*}x & = &7.62835577107986... \\\alpha & = & 0.805235953662952... \\& = & 46.1366216570791... \textrm{(degrees)}\end{eqnarray*}

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