Law of Sines 7.1 JMerrill, 2009

A Generic Triangle B c a C A b

Solve the right triangle.

Solve the right triangle.

Oblique Triangles But what if the triangle isn’t right? We need a method that will work for what we call oblique triangles. (Triangles that aren’t right)

Law of Sines B C A Usually used when you have ASA or AAS In ANY triangle ABC: (Notice that each angle goes with its corresponding side in each proportion. One of the proportions must have a “known” angle and side) c b B a C

Example: Find the missing variable In the triangle below, mA = 33o, mB = 47o & b = 14, find a. A You know both B’s, the angle and the side, so that will be our “known” B 33 47 14 C

Example 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 = 25m,  A = 110o, and B = 20o. Find AC and BC. C A B

Example Mark the drawing with known info C A B 3. 4. 25 20o B

You Do Find ALL missing angles and sides in the triangle below: mA = 28o, a = 12 & b =24 C A mB = 70o mC = 82o C = 25.312 B

Ambiguous Case When two sides and a non-included angle are given, there are several situations possible – this is called the ambiguous case. There could be only one triangle. There could be two triangles (when an angle(s) has more than one possibility). There could be no triangle (when you take the inverse sine of a value larger than 1).

Example Find angle B in triangle ABC if a = 2, b = 6, and A = 30o Applying the Law of Sines, we have : Since sinB can never be larger than 1, this triangle does not exist.

Example 2 Find the missing parts in triangle ABC if a = 54cm, b = 62cm, and A = 40o First solve for B Since B can be 48o or 132o, C can be 92o or 8o Then c can be 84cm or 12cm (using Law of Sines) Thus, there are 2 possibilities and no way to tell which is correct—ambiguous!

Example 3-Application A forest ranger at an observation point (A) sights a fire in the direction 32° east of north. Another ranger at a second observation point (B), 10 miles due east of A, sights the same fire 48° west of north. Find the distance from each observation point to the fire. 8.611 80o 6.795 32o 48o 42o 58o 10 B A