Expert: Paul Klarreich - 11/5/2006

Question
Dear Paul,
My problem involves two composite triangles, PQR (smaller) and PQZ (larger), with the following dimensions: side PQ = 25, side PZ = 66, side QR = 25, angle Z = 10.5 degrees. How would I solve for angle QPR/QPZ and angle QRZ?

Answer
Questioner:  Maya
Category:  Advanced Math
Private:  no
 
Subject:  trigonometry: sine law/ cosine law
Question:  Dear Paul,
My problem involves two composite triangles, PQR (smaller) and PQZ (larger), with the following dimensions:

side PQ = 25, side PZ = 66, side QR = 25, angle Z = 10.5 degrees. How would I solve for angle QPR/QPZ and angle QRZ?
...........................................
Hi, Maya,

I think I have your diagram right.  You didn't say that P,R,Z are in a straight line, but that seems clear from other things.

Start by looking at triangle PQZ - the larger one - and draw separately so you don't get distracted by the other elements.  In triangle PQZ, you have:

PQ = 25 = z   (remember the convention: side small z is opposite angle big Z)
angle Z = 10.5 deg
PZ = 66 = q
angle Q is not known.

Now use the Law of Sines:

sin Q   sin Z
----- = -----
 q       z


sin Q   sin 10.5
----- = --------
 66       25
       66 sin 10.5
sin Q = ------------
           25

You will have to use your calculator now. (sorry - no escaping this)  My Windows calculator gives:

sin Q = 0.48110178729926928543079370059476

and

Q1 = 28.757386229684925735958259808524 degrees.

Now there is a difficulty, because the triangle situation is S-S-A, which you may remember is the 'ambiguous case'.  

[The S-S-A code comes from your high school geometry, where you proved triangles congruent by several methods.  You wrote, for example,

A-S-A, to mean you had two angles and the side between them.  

S-A-S meant two sides and the angle between them.  

But S-S-A means two sides and an angle that is NOT between them.]

That means there could be two solutions.  They are:

Q1 = 28.757386229684925735958259808524 degrees.

and

Q2 = 180 - 28.757386229684925735958259808524
  = 151.24261377031507426404174019148

Now which is correct in this case?  I think you will see that if we take the smaller value, Q1, then PQZ becomes the smaller triangle, and that is wrong.  So our preliminary solution is that

(we can discard most of those digits now)

Q = 151.24 degrees.

Then we find that

Angle P = 180 - (151.24 + 10.5) = 180 - 161.74 = 18.26 degrees.

and the rest of the angles can be deduced easily.