Question

1.) The trig ratio needed to solve the problem and how that it is the correct trig function.

Answer 1

In every right triangle, you have the following identites:

$o = h\sin(\alpha),\quad a = h\cos(\alpha)$

Where $\alpha$ is one of the acute angles, $o$ is the leg opposite to the angle, $a$ is the leg adjacent to the angle and $a$ is the hypothenuse.

In this case, we know that the adjacent leg is 100ft long, so we can use the second formula to compute the hypothenuse:

$100 = h\cos(37) \implies h = \dfrac{100}{\cos(37)}$

Now let's use the first equation to compute the length of the opposite leg, i.e. the one you're interested in:

$o = h\sin(\alpha) = \dfrac{100}{\cos(37)}\sin(37)$

Note that the ratio between the sine and the cosine is the tangent:

$o = 100\tan(37)$

If you ask for the tangent of 37 to a calculator, you get

$\tan(37) \approx 0.753554050\ldots$

So, you have

$100\tan(37) \approx 75.3554050\ldots$

Which rounded to the nearest tenth is 75.36