Quandaries
and Queries 

Who is asking: Parent Level: All Question: Given 1. B = 20 Degrees, a = 25, b = 16 

Hi Allen, There are two identities in trigonometry that are very useful in these problems. They are the law of sines and the law of cosines. The law of sines is the easier to work with so I always look first to see if I can use is. It says
This identity involves four quantities, the side lengths a and b and the angle measurements A and B. What this means is that if you know any three of these quantities you can find the fourth. In particular if you know an angle measurement, the length of the side opposite and either another angle or another side you can find the fourth. This is the situation in problem 3. You know the angle measurement C, the side opposite c and the angle A so you can find the side length a. (I wrote the law of sines using A,a,B and b but I could as easily used A, a, C and c.)
Solve for a. Since the sum of the angles in a triangle is 180 degrees
and thus you can use the law of sines again to find b. Problem 1 is similar. You know B, b, and a so you can use the law of sines to find sin(A) and then A. For problem 2 the only angle you know is A and you don't know a so you can't use the law of sines. The law of cosines states
To me this says, "if you know the size of an angle (here called C) and the lengths of the two sides that are not opposite C then you can find the length of c." This is the situation in problem 2. You know A as well as b and c so you can find a. That is
Solve for a. Now you know A, a and b so you can use the law of sines to find B. Harley 

