Law of Sines Lesson 6.4

The Law of Sines Let’s now solve oblique triangles (Δ’s without a right angle): B A C a c b h

When to Use the Law of Sines When we know one side and two angles (ASA or SAA) When we know two sides and an angle opposite one of those sides (SSA) Basically, we need to know a side and the angle opposite that side C = 112° a b = 216.75 B 44.5° c A =23.5°

Using the Sine Law We know 2 sides and angle opposite one of the sides: Now how would we find angle C and then side c? C a =9.5 b=15 A c B = 47°

Example 1: A satellite orbiting the earth passes directly overhead and between Phoenix and L.A., 340 mi apart. The angle of elevation is simultaneously observed to be 60° at Phoenix and 75° at L.A. How far is the satellite from L.A.? 75° 60° 340 mi

Ex 1 Solution: How far is the satellite from L.A.? 45° c b 75° 60° LA 340 mi PHO

Height of a Kite Two observers directly under the string and 30' from each other observe a kite at an angle of 62° and 78°. How high is the kite? h 62° 78° 30

Height of a Kite (cont’d) 62° 78° 30

The Ambiguous Case (SSA) Given two sides and an angle opposite one of them, several possibilities exist: No solution, side too short to make a triangle One solution, side equals altitude 2 10 20° 20° 10 3.42

The Ambiguous Case (SSA) Two solutions: 2 triangles are possible (why?) One unique solution, the opposite side is longer than adjacent side Solving for A could give either an acute or obtuse angle! 10 5 5 20° A A' 13.42 10 20°