# How do you apply trigonometric equations to solve real life problems?

Apr 27, 2018

When at $\left({71.06047}^{\circ} W , {43.08350}^{\circ} N\right)$ a distant cell tower is at heading ${131}^{\circ} \left(S E\right)$, and at $\left({71.06137}^{\circ} W , {43.08007}^{\circ} N\right)$ it's at heading ${99}^{\circ} \left(E\right)$. Where is the tower?

#### Explanation:

Someone from San Antonio requested an answer three years ago! Hope they've figured it out by now.

Here some real life trig I've been meaning to do. I wanted to know where that cell phone tower I can see from my house is. It's on a hill in the distance.

From one spot near my house, maybe the one in this picture, I pointed my phone at the tower with my GPS app on and got this:

The relevant information is:

$\left({71.06047}^{\circ} W , {43.08350}^{\circ} N\right)$ heading ${131}^{\circ} \left(S E\right)$

At another spot I got

$\left({71.06137}^{\circ} W , {43.08007}^{\circ} N\right)$ heading ${99}^{\circ} \left(E\right)$

The heading is relative to due north. Here's a figure.

We treat west as negative. I translated the origin to (-71.06, 43,08) and multiplied the coordinates by 100,000 so our new problem is:

$\left(- 047 , 350\right)$ heading ${131}^{\circ} \left(S E\right)$

$\left(- 137 , 007\right)$ heading ${99}^{\circ} \left(E\right)$

Find Tower coordinates $\left(x , y\right)$

To solve we write the equations of the two lines and find the meet.

When measured relative to the y axis like that, the heading angle $\theta$ converts to a slope as $m = \cot \theta$.

So our two lines

$y - 350 = \left(x + 47\right) \cot 131$

$\left(y - 7\right) = \left(x + 137\right) \cot 99$

meet when

 350 + (x+47) cot 131 = 7+ (x+137) cot 99 

$x = \frac{7 + 137 \cot 99 - 350 - 47 \cot 131}{\cot 131 - \cot 99}$ x

 x≈ 455.537

y ≈-86.849 #

That means my tower is at GPS coordinates

$\left(- 71.06 + .00456 , 43.08 - .000868\right) = \left(- 71.0554 , 43.07132\right)$

Let's check Google Maps. Pretty good, off by around 400 feet.