## Finding theta in terms of h and l of a right triangle

Trigonometric ratios and functions, the unit circle, inverse trig functions, identities, trig graphs, etc.
johnbean
Posts: 1
Joined: Sat May 21, 2011 1:11 am
Contact:

### Finding theta in terms of h and l of a right triangle

Hi, first post, but need help quick. I am writing a program in visual basic (irrelevant) and need to figure out a theta value from a right triangle, but need it in terms of certain variables. The figure is shown below with variables theta, l, and h. I need the theta value in terms of h and l.

So far I got theta = sin-1( sqrt(h*l)/l), however, this wasn't correct.

Another approach I took led me to sin(theta) tan(theta) = h/l . But I am unsure how to get theta isolated.

Any help is appreciated.

Thanks,

John

nona.m.nona
Posts: 288
Joined: Sun Dec 14, 2008 11:07 pm
Contact:

### Re: Finding theta in terms of h and l of a right triangle

...I need the theta value in terms of h and l.....

For clarity, I will use upper-case "L" for the "length" of the longer side of the original (largest) right triangle.

If the remaining portion of the original triangle's hypotenuse is labeled as "x", then the hypotenuse has length x + h. The remaining angle may be labelled $\beta$. It is elementary to prove that one then has three right triangles:

largest: angle $\theta$ contained by sides L and x + h
middle: angle $\theta$ contained by sides L and x
smallest: angle $\theta$ contained by sides $\sqrt{(x\, +\, h)^2\, -\, L}$ and $\sqrt{L^2\, -\, x^2}$

From the largest and middle triangles, one may arrive at two expressions for the cosine of $\theta$ which may be used to create an equation. Solve this "quadratic" for the "value" of x in terms of L and h by applying the Quadratic Formula.