9.3 The Law of Sines

9.3/9.4 Laws of Sines and Cosines Objectives: 1. Solve non-right triangles. Vocabulary: Law of Sines, Law of Cosines

Derivation of the Law of Sines sinAsinC

Derivation of the Law of Sines Is the same true if we start with an obtuse triangle?. sinAsinC When  A is obtuse, then noting that sin(  – A) = sinA  – A

AB C a b c The Law of Sines Alternative forms are sometimes convenient to use:

Using the Law of Sines in an Application (ASA) Two stations are on an east-west line 110 miles apart. A forest fire is located on a bearing of N 42 o E from the western station at A and a bearing of N 15 o E from the eastern station at B. How far is the fire from the western station?  A = 90 o – 42 o = 48 o  B = 90 o + 15 o = 105 o  C = 180 o – 105 o – 48 o = 27 o Using the law of sines to find b gives

Then by the law of sines: A civil engineer wants to determine the distances from points A and B to an inaccessible point C, as shown. From direct measurement the engineer knows that AB = 25 m,  A = 110 o, and  B = 20 o. Find AC and BC.