1. 5.6 The law of sines
Solve triangles by using the Law of Sines if the measures of two angles and a side are given.
Find the area of a triangle if the measures of two sides and the included angles or the measures of two angles and a side are given.

2. Law of Sines Law of Sines
How can you find lengths & angles of any triangle?
Law of Sines
Let ΔABC be any triangle with a, b, and c representing the measures of the sides opposite other angles with measures A, B, and C, respectively. Then, the following is true. 𝒂 𝐬𝐢𝐧 𝑨 = 𝒃 𝐬𝐢𝐧 𝑩 = 𝒄 𝐬𝐢𝐧 𝑪
When using law of sines, triangle information ASA and AAS gives a unique solution.

3. Example
1) Solve ΔLMN if L = 29°, M = 112° and l = 22 2) Solve ΔABC if A = 40º, B = 59º and c = 14 L M N n l m

4. Area of Triangles Area of Triangle = 1 2 𝑏ℎ
Where b – is the base and h – is the height of the triangle.
Other formulas to find the area of a triangle
Let ΔABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, and C, respectively. Then the area K can be determined using one of the following formulas.
𝐾= 1 2 𝑏𝑐 sin 𝐴 𝐾= 1 2 𝑎𝑐 sin 𝐵 𝐾= 1 2 𝑎𝑏 sin 𝐶
𝐾= 1 2 𝑎 2 sin 𝐵 sin 𝐶 sin 𝐴 𝐾= 1 2 𝑏 2 sin 𝐴 sin 𝐶 sin 𝐵 𝐾= 1 2 𝑐 2 sin 𝐴 sin 𝐵 sin 𝐶

5. Examples
1) Find the area of ΔABC if b = 21.2, c = 16.5 and A = 25° 2) Find the area of ΔJKL if j = 45.7, K = 111.1°and L = 27.3° A B C c a b J L K k l j

6. Word Problem
A hot air balloon is flying above Gorveburg. To the left side of the balloon, the balloonist measures the angle of depression to the Groveburg soccer fields to be 20º15’. To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be 62º30’. The distance between the two athletic complexes is 4 miles.
Find the distance from the balloon to the soccer fields.
Find the distance from the balloon to the football field.