How Do You Calculate Distance Using Trigonometry in Physics?

[a_j]

Ok. This is a fairly simple question since it is problem #11 of Chapter 1! Yet, I cannot do it. (Hopefully this isn't a sign of things to come).

Anyway, basically the problem goes like this...

A monument rises to a height of 192m. You estimate the line of sight with the top of the arch to be 2.0 degrees above the horizontal. Apporximately how far (in km) are you from the base of the arch?

The answer is 5.5km, but I cannot get this. I can't figure out which inverse trig funtion to use. The closest I came was 6.7.

Please explain which trig function and why its that function. I tried the tangent of 2.0 and a few other methods and haven't came up with anything.

Thanks in advance,
a_j

 

[jamesrc]

Draw the triangle out and it should help you solve the problem. Looking from the side, the height of the triangle is 192 m (this is the distance between the ground and the top of the arch). The base of the triangle is the unknown you are trying to solve for (the distance you are away from the arch). The angle is given as 2 degrees. Tangent is a good choice to solve this problem, but how did you set it up? The tangent of the angle is equal to the ration of the opposite side to the adjacent side. So, if we call the unknown distance d:

\tan{2^\circ} = \frac{194{\rm m}}{d}

and you solve for d. (Notice you don't need to take an inverse trig function in this problem.)

 

[M98Ranger]

The trig function you would use in this problem is the tangent function. This is because the tangent function relates the opposite side of a right triangle to the adjacent side, which is exactly what we need to find in this problem. The opposite side is the height of the monument (192m) and the adjacent side is the distance from the base of the arch to your line of sight. We can set up a tangent ratio using these values:

tan(2.0 degrees) = opposite/adjacent
tan(2.0 degrees) = 192m/x

To solve for x, we can rearrange the equation and use a calculator to find the value of x:

x = 192m/tan(2.0 degrees)
x = 192m/0.0349
x = 5501.4m

Since we want the answer in kilometers, we can convert this to kilometers by dividing by 1000:

x = 5.5km

So your estimated distance from the base of the arch is 5.5km. It is important to use the tangent function in this problem because it directly relates the height of the monument to the distance we are trying to find. Keep practicing and you'll become more comfortable with these types of problems!