In this section we will solve find all the sides and angles of oblique triangles triangles that have no right angles As standard notation the angles of a triangle are labeled ID: 654222
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Slide1
19.
Law of
SinesSlide2
Introduction
In this section, we will solve (find all
the sides and angles of) oblique triangles – triangles that have no right angles.As standard notation, the angles of a triangle are labeled A, B, and C, and their opposite sides are labeled a, b, and c.To solve an oblique triangle, we need to know the measure of at least one side and any two other measures of the triangle—either two sides, two angles, or one angle and one side.Slide3
Remember ……Slide4
4 cases
Two
angles and any side are known (AAS or ASA)Two sides and an angle opposite one of them are known (SSA)Three sides are known (SSS)Two sides and their included angle are known (SAS)The first two cases can be solved using the Law of Sines
,
the
last two cases require the
Law of Cosines
.Slide5
CASE 1: ASA or
SAA Law of
sines
S
A
A
ASA
S
A
A
SAASlide6
S
S
A
CASE 2:
SSA Law of
sinesSlide7
S
S
A
CASE 3:
SAS Law of cosinesSlide8
S
S
S
CASE 4:
SSS Law of cosinesSlide9
Law of Sines
A
B
C
a
b
cSlide10
Case 1 - AAS
For the triangle below
C = 102, B = 29, and b = 28 feet. Find the remaining angle and sides.Slide11
Example AAS -
Solution
The third angle of the triangle is A = 180 – B – C = 180 – 29 – 102 = 49.
By the Law of Sines, you have
.Slide12
Example AAS –
Solution
Using b = 28 producesand
cont’dSlide13
Case 1 -
ASA
A
B
C
c
a
b
C=70
o
b
=44.1
a=32.7Slide14
Example
A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is
Draw a diagram
Draw Then find theSlide15
Example
cont
Cross products
Use a calculator.
Law of Sines
Answer:
The length of the shadow is about 75.9 feet.
Divide each side by sin
Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow.Slide16
Example
A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water?
Answer:
About 42 feet of the fishing line that is cast out is above the surface of the water.Slide17
Area of an Oblique
Triangle (SAS)Slide18
Area of a
Triangle
A
B
C
c
a
b
h
Area
= ½
ab
(sin C) = ½ ac(sin B) = ½
bc
(sin A) Slide19
Example –
Finding the Area of a Triangular Lot
Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of 102.Solution:Consider a = 90 meters, b = 52 meters, and the included angle C = 102Then, the area of the triangle is Area = ½ ab sin C = ½ (90)(52)(sin102
)
2289 square meters.Slide20
20
Try It Out
Determine the area of these triangles
127°
12
24
76.3°
42.8°
17.9