6.1 Law of Sines Objectives –Use the Law of Sines to solve oblique triangles –Use the Law of Sines to solve, is possible, the triangle or triangles in the ambiguous case –Find the area of an oblique triangle using the sine function –Solve applied problems using the Law of Sines Pg. 635 #2-42 (every other even), and evens

Law of Sines Previously, our relationships between sides of a triangle and the angles were unique only to RIGHT triangles What about other triangles? Any triangle that is NOT a right triangle is called Oblique. Oblique triangles have either three acute angles or two acute angles with one obtuse angle The following relationship exists (A,B,C are measures of the 3 angles; a,b,c are the lengths of sides opposite those angles): Note: The law of sines is applied using equivalent fractions. So, only two of these ratios are used at one time (NOT all three)

Solving an oblique triangle 1. If given: A = 50 degrees, B = 30 degrees, b = 7 cm. Can you solve this triangle? You know C = 100 degrees. You can find “a” & “c” by law of sines. Note: All diagrams will NOT be in proportion to the correct angles and sides.

2. Solve a triangle with A = 64 o, C = 33.5 o, c = 14 centimeters. Round answers to the nearest tenth.

3. Solve triangle ABC if A = 40 o, C = 22.5 o, and b = 12 units. Round answers to the nearest tenth.

It is important to keep the following relationship in mind. With all triangles: The shortest side is always opposite the shortest angle. Likewise, the longest side is always opposite the longest angle. It may be hard to resist the urge to draw triangles that are in proportion from the beginning of the solving process. However, certain assumptions about the triangle must be made to make these sketches. In some cases, you will not know if you have three acute angles in your triangle or two acute paired with one obtuse. For this reason, we will continue to use diagrams that are generic (not in proportion to the given data). When you cannot determine the types of angles the triangle has from the given data, we call this an ambiguous case. It occurs when only one angle is given. For ambiguous triangles, you will find one solution, no solution, or two solutions.

4. Solve triangle ABC if A = 57 o, a = 33, and b = 26. Round to the tenths place.

5. Solve the triangle ABC if A = 50 o, a = 10, and b = 20. Round to the tenths place.

6. Solve triangle ABC if A = 35 o, a = 12, and b = 16. Round to the tenths place.

Finding the area of an oblique triangle Area= ½ b c sinA Area= ½ a b sinC Area= ½ a c sinB 7. Find the area of a triangle having two sides of lengths 8 meters and 12 meters and an included (in between) angle of 135 o.