Friday, October 5, 2012

Here's an application of the Law of Sines. The Law of Sines is used to solve for the sides and angles of a triangle which is not a right triangle.

Law of Sines:

Sine A/a = Sine B/b = Sine C/c

Example:

Suppose a man is 100 feet from a building.  The angle of elevation from the man's feet to the top of the building is 41 degrees. The angle of elevation to the top of a poster on the wall is 21 degrees. Assume the side of the building is perpendicular to the ground. How far is it from the top of the building to the top of the poster?

It's easiest to draw a diagram, which you'll see below.

We can use the Law of Sines to get the height of the building. We'll make the height of the building "x"

x/Sine 41 = 100/Sine 49

Using a calculator to find the values for Sine 41 and Sine 49 we get

x/0.656 = 100/0.7547

0.7547x = 65.6

x = 86.92 feet

Now we make the height from the ground to the poster, x  and from the poster to the top of the building 86.92 - x.  We do this because the total height must be 86.92 and 86.92 - x + x = 86.92 (the -x and +x cancel)

Notice how we label the diagram knowing the fact that the height of the building is 86.92 feet.

Now we can solve for x since we know the lower triangle has angle measures 69, 90 and 21. See the diagram below.

We can solve for x as follows:

x/Sine 21 = 100/Sine 69

x/0.3584 = 100/0.9336

0.9336x = 35.84

x = 38.39

So, it's 38.39 feet from the ground to the poster. Therefore it's 86.92 - 38.39 = 48.53 feet from the top of the building to the top of the poster.