1. WARM UP -0.372… scalene oblique
State the law of cosines for ΔPAF involving angle P.
State the formula for area of the ΔPAF involving angle P.
If sin θ= 0.372…, then sin (-θ) =
An _____________ triangle has no equal sides and no equal angles.
An ______________ triangle has no right triangle.
-0.372…
scalene
oblique

2. OBLIQUE TRIANGLES: LAW OF SINES

3. OBJECTIVES
Given the measure of an angle, the length of the side opposite this angle , and one other piece of information about a triangle, find the other side lengths.
Key term: law of sines

4. INTRODUCTION
Because the law of cosines involves all three sides of a triangle, you must know at least two of the sides to use it. In this section, the law of sines, which lets you calculate a side length of a triangle if only one side and two angles are given.
The figure shows ΔABC. Previously you learned that the area is equal to ½ bc sin A. The area is constant no matter which pair of sides and included angle you use.
Set the areas equal.
Multiply by 2.

5. PROPERTY: THE LAW OF SINES
The final relationship is called the law of sines. If three nonzero numbers are equal then their reciprocals are equal. So you can write the law of sines in another algebraic form:
In ΔABC,
and
Verbally: within any given triangle, the ratio of the sine of an angle to the length of the side opposite that angle is constant.

6. PROPERTY: THE LAW OF SINES
Because of the different combinations of sides and angles for any given triangle, it is convenient to revive some terminology from geometry. The acronym SAS stands for “side, angle, side.” This means that as you go around the perimeter of the triangle, you are given the length of a side, the measure of an angle and the length of a side, in that order. So SAS is equivalent to knowing two sides and the included angle, the same information that is used in the law of cosines and in the area formula. Similar meanings are attached to ASA, AAS, SSA, and SSS.

7. GIVEN AAS, FIND THE OTHER SIDES
Example 1 shows you how to calculate two side lengths given the third side and two angles.
Example 1: In ΔABC, B = 64°, C = 38°, and b = 9 ft. Find the lengths of sides a and c
Case: AAS
Solution: First draw a picture.
Because you know the angle opposite side c but not the angle opposite side a, it’s easier to start with finding side c
Use the appropriate parts of the law of sines. Put the unknown in the numerator on the left side.
Multiply both sides by sin 38° to isolate c on the left.
c = …

8. GIVEN AAS, FIND THE OTHER SIDES
To find the law of sines, you need the measure of A, the opposite angle.
A = 180° – (38° + 65°) = 78°
The sum of the interior angles in a triangle is 180°.
Use the appropriate parts of the law of sines with a in the numerator.
a = …
a ≈ 9.79 ft. and c ≈ 6.16 ft.

9. GIVEN ASA, FIND THE OTHER SIDES
Example 2 shows you how to calculate side lengths if the given side is included between the two given angles.
Example 2: In ΔABC, a = 8 m, B = 64°, and C = 38° Find the lengths of sides b and c
Solution: First draw a picture. The picture reveals that in this case you do not know the angle opposite the given side. So you calculate this angle first. From there on, it is a familiar problem, similar to Example 1.
Case: ASA
A = 180° – (38° C 65°) = 78°
Use the appropriate parts of the law of sines.
Use the appropriate parts of the law of sines.
b ≈ 7.35 m and c ≈ 5.04 m

10. LAW OF SINES FOR ANGLES
You can use the law of sines to find an unknown angle of a triangle. However, you must be careful because there are two values of the inverse sine relation between 0° and 180°, either of which could be the answer. For instance, arcsin 0.8 = …° or …°, both could be angles of a triangle. Problem 11 in your homework will show you how to handle this situation.

11. SUMMARIZATION WHAT YOU ARE GIVEN WHAT YOU WANT TO FIND LAW TO APPLY
Three side (SSS)
An unknown angle
Law of cosines
Two sides and an included angle (SAS)
The unknown side
Two angles and an included side (ASA)
An unknown side
Law of sines (must first find missing angle, using 180° - (A – B))
The unknown angle
180° - (A + B)
Two angles and a nonincluded side (AAS)
Law of sines
180° - ( A + B)

12. Ch. 6.4 Homework
Textbook pg. 262 #1-10 all
Extra credit pg. 263 #14