·1. what does it mean to solve a right triangle? · 2. how can you solve right triangles using similarity? the pythagorean theorem? · 3. what is the relationship between the sides and angle measures of right triangles that describe trigonometric functions? · 4. what are the inverse trigonometric functions? how are the inverse functions used when solving right triangles? · 5. what is the relationship between the angle of elevation and the angle of depression? give an example of an application using the angle of elevation or the angle of depression.

1. WHAT DOES IT MEAN TO SOLVE A RIGHT TRIANGLE?  Every triangle has 3 angles and 3 sides.  A right triangle has one 90-degree angle and a hypotenuse, the side opposite the right triangle.

To "solve a right triangle," we must come up with the length of each of the 3 sides and the measures of the 2 angles beyond the 1 right triangle.

Example:  Given the right triangle with one of the smaller angles equal to 30 degrees and hypotenuse equal to 7, "solve" the triangle.  In other words, find the lengths of the 2 legs and the measure of the 3rd angle.  That angle is 90-30, or 60 degrees.  Thus, you have a 30-60-90 triangle.

One way to "solve" this triangle is to use the Law of Sines:

  a            b              c
= =
sin A      sin B       sin C.

If we let c = 7 and C = 90 deg, then the side length opposite the 30 degree angle is 

    a             7
=
sin 30     sin 90

and this simplifies to a(1) = (7)(sin 30) = 7/2

So now we know that one angle is 90 degrees, one is 30 and one is 60.

The 7 inch hypotenuse is opposite the 90 degree angle; the side opposite the 30 degree angle is 7/2, and the side opp. the 60 degree angle is 7sqrt(3)/2.