We have the Law of Sines,

which is A over sin A is equal to B over sin B is equal to C over sin C.

In other words, the length of this side A divided by

the sin of the angle which is opposite to us.

Is equal to the length of b divided by the sin of its opposite angle, etcetera.

And another convenient law or equation is the law of cosines,

which can be written in terms of the three different angles as shown here.

A squared is equal to b squared plus c squared minus 2bc cosin a.

And similar equations for the other two angles.

The sum of the interior angles in a triangle is equal to 180 degrees,

and more general, for polygons of arbitrary number of sides,

the sum of the interior angles Is equal to the number of sides or

the number of angles, minus two, multiplied by 180 degrees.

And for regular polygons, such as sketched here where all the angles are equal,

and all the lengths of the sides are equal,

the magnitude of the interior angles is number of sides minus two,

times 180 divided by the number of sides or angles, N.

Let’s do an example on that.

And the question is an observer surveys a building and

observes that the vertical to its top is 40 degrees.

He then walks 50 meters farther away and

observes the vertical angle to be now 30 degrees.

The height of the building is most nearly which of these alternatives?

So, here's a sketch.

So we have the building and here is the first observation point which are labelled

A and he observes at the angle to the top of the building 40 degrees.

Then he walks 50 metres away so this distance is 50 metres To the point b,

where he observes that the angle is 30 degrees, and

the question is, how high is the building?

In other words, the height CD in those triangles.

So the steps involved here, first we want to find this angle, theta.

By looking at the triangle ADC, the right angle triangle.

So the sum of the angles in that triangle is 180 degrees,

so 40 degrees plus 90 degrees plus theta is 180.

Therefore, theta is 50 degrees.

Next we want to find the Angle phi by observing the triangle BDC.

And again, the sum of the angles in that triangle is equal to 180 degrees,

therefore 30, plus the right angle, plus phi plus theta.

Is 180.

And we already know theta, so

therefore the angle phi is ten degrees.

Next we'll use the law of sines on the triangle ABCABC

to find the length of the inclined side AC.