Download PC 6-1 Law of Sines

Chapter 6
Additional Topics in Trigonometry
6.1 The Law of Sines
Objectives:
Use Law of Sines to solve oblique triangles
(AAS or ASA).
Use Law of Sines to solve oblique triangles
(SSA).
Find areas of oblique triangles.
Use Law of Sines to model & solve real-life
problems.
2
Oblique Triangles
Have no right angles.
Angles labeled with
capital letters.
Sides labeled with lowercase letters (same letter as
opposite angle).
3
Proof of Law of Sines
Let h be the altitude of the triangle.
Find sin A and sin B.
4
Proof continued….
Solve each equation for h.
Set h = h.
5
The Law of Sines
6
Example 1
For
ABC, C = 102.3°, B = 28.7°, and b = 27.4
feet. Find the remaining angle and sides.
7
Example 2
A pole tilts toward the
sun at an 8° angle from
the vertical, and it casts
a 22-foot shadow. The
angle of elevation from
the tip of the shadow to
the top of the pole is 43°.
How tall is the pole?
8
Possible Combinations for
Law of Sines
 Given a, b, A, and where h = b sin A:
A is acute
a=h
a>b
A is obtuse
a<h
a≤b
h<a<b
a>b
 We will look at each one individually.
9
A is Acute and a < h
No triangle formed - a can’t reach the base.
10
A is Acute and a = h
One triangle is formed.
11
A is Acute and a > b
One triangle is formed – a intersects the
base at only one point.
12
A is Acute and h < a < b
(Ambiguous Case)
Two unique triangles can be formed.
Side a can intersect the base at 2 points.
13
A is Obtuse and a ≤ b
No triangle is formed – a can’t reach the base.
14
A is Obtuse and a > b
One triangle is formed.
15
The Ambiguous Case
Given A, a, and b, we can find h
h = b sin A
Is a > b? If so, only 1 triangle.
Is h < a < b? If so, then 2 triangles.
16
Example 3
For
ABC, a = 22 inches, b = 12 inches, and
A = 42°. Find the remaining side and angles.
17
Example 4
For
ABC, a = 12 meters, b = 31 meters,
and A = 20.5°.
a. How many triangles can be formed?
b. Find all remaining side(s) and angles.
18
Example 5
Show that there is no triangle for which
a = 15, b = 25, and A = 85°.
19
Area of an Oblique Triangle
The area of any triangle is one-half the
product of the lengths of two sides times
the sine of their included angle.
What happens if the included angle is 90°?
20
Example 6
Find the area of a triangular lot having
two sides of lengths 90 meters and 52
meters and an included angle of 102°.
21
Homework 6.1
Worksheet 6.1
22