Download So in triangle ABC

Pre-Calculus
Lesson 20: The Law of Sines
Name: ___________________________
Date: _____________
Period: 1 2 3 4 5 6 7 8
Trig functions can be used to solve right triangles, much like we saw with the past few
lessons. However, they can also be used to solve what are called oblique triangles,
triangles with no right angles. To do this, we will study the Law of Sines and Law of
Cosines. To state these laws more easily, we follow the convention of labeling the angles
of a triangle A, B, C and the lengths of the corresponding opposite sides as a, b, c.
In order to solve a triangle, we need to know certain information about its sides and
angles. To see what information we have, we should always make a sketch of the
triangle. We may be given any combination of: ASA, SAA, SSA, SAS or SSS.
In general, the four cases below are created based on knowing 3 of a triangles 6 parts as long as at least one of the these 3
parts is a side.
The Law of Sines
In any triangle, the lengths of the sides are proportional to the values of the sine of the corresponding opposite angles.
So in triangle ABC,


or


Examples
1. [Case 1: ASA] A satellite orbiting the earth passes directly overhead at observation stations at Phoenix and LA,
340 miles apart. At an instant when the satellite is between these two stations, its angle of elevation is
simultaneously observed to be 60° at Phoenix and 75° at LA. How far is the satellite from LA?
Pre-Calculus
Lesson 20: The Law of Sines
2.
Name: ___________________________
Date: _____________
Period: 1 2 3 4 5 6 7 8
[Case 1: SAA] Solve the following triangle.
The Ambiguous Case (Case 2)
In examples 1 & 2, one unique triangle was determined based on the information given. But in Case 2 triangles (SSA) there
may be two possible triangles, one triangle, or no triangle with the given properties. Hence, Case 2 often gets the infamous
name the “ambiguous case”. For example, if sides a and b are given along with angle A, look at the following possibilities:
In (a) no solution is possible, since side a is too short to complete the triangle. In (b) the solution is a right triangle. In (c)
two solutions are possible, and in (d) there is only one unique triangle.
Examples
3.
[Case 2: SSA, 1 Solution] Solve triangle ABC, where
.
*In this example, there were two possibilities for angle B, and one of them wasn’t compatible with the rest of the
information. In general, if the sin A < 1, we must check the angle and its supplement as possibilities, because any angle
smaller than 180° can be in the triangle.
Pre-Calculus
Lesson 20: The Law of Sines
4.
[Case 2: SSA, 2 Solutions] Solve triangle ABC, where
5.
[Case 2: SSA, No Solutions] Solve triangle ABC, where
Name: ___________________________
Date: _____________
Period: 1 2 3 4 5 6 7 8
.
.
Pre-Calculus
Lesson 20: The Law of Sines
Name: ___________________________
Date: _____________
Period: 1 2 3 4 5 6 7 8
We can also find the area of an oblique triangle if we have two sides and an included angle. This is the same scenario as
Case 3 from above. If the included angle is obtuse, simply use it’s reference angle.
A =
Example
6. Find the area of triangle ABC below.
1
1
1
bcsin A = acsin B = absinC
2
2
2