Download Oblique triangle

December 11, 2014
!!!
!
G
I
R
T
Yeah
You are interested in buying a triangular plot of land in Michigan
and would like to know how much it would cost to fence it in. You
spilled your beverage on the realtor's map and messed it up. All
you can read are two interior angles 95° and 68°and 115 ft. for
the side between these angles. What are the lengths of the
other two sides?
115
?
95°
68°
?
Chapter 6 Additional Topics in Trigonometry
Section 6.1 Law of Sines
Objective: Use the Law of Sines to solve oblique triangles and find the
areas of oblique triangles.
Oblique triangle - A triangle that has no right angles.
A
Acute Triangle
b
a
h
c
h
C
c
i s tr
lve th
o
s
u
o
y
le.)
Could
to sca
rawn
(Not d
Obtuse triangle
b
?
iangle
A
8
B
B
a
?
5
?
30°
?
C
December 11, 2014
Note: To solve an oblique triangle, you need to know at least one
side and then any two other parts of the triangle.
So, 4 cases...
1) 2 angles and any side (AAS or ASA)
Use
Law of Sines
2) 2 sides and one opposite angle (SSA)
Use
Law of Cosines
3) Three sides (SSS)
4) 2 sides and their included angle (SAS)
Law of Sines
C
b
A
b
c
a
=
=
sin A sin B sin C
A
Acute Triangle
c
a
h
c
or
h
B
Obtuse triangle
b
C
B
a
sin A sin B sin C
a = b = c
"The Ambiguous Case" (SSA)
➔3 possibilities: no triangle exists, one, or two
➔ calculate the height (h = b sin A) so you can
compare a, h, and b.(see p.394)
A
8
B
Drawn to
scale.
?
5
?
30°
?
C
December 11, 2014
h = Adj. Side sin (Given Angle)
p. 394
Given opp. side
Given adj. side
S
A
h
Given Angle
S
If Given Angle is acute...
Opp. Side < h ------------> none
Opp. Side = h ------------> One (rt.
Opp. Side > Adj. side -----> One
h < Opp. Side < Adj. Side ---> Two
)
If Given Angle is obtuse...
Opp. Side ≤ Adj. Side -----> none
Opp. Side > Adj. Side -----> One
How many possible triangles?
1. A = 62 , a = 10, b = 12
2. A = 98 , a = 10, b = 3
3. A = 54 , a = 7, b = 10
Ex1: Given B = 78, c = 12, and b = 5, find angle C.
none
1
none
December 11, 2014
Ex2: Given A = 123°, B = 41°, and a =10 inches, find c.
!
S i n es
f
o
Law
e
h
t
e
Us
Ex3: A triangular plot of land has interior angles A = 95° and
C = 68°. If the side between these angles is 115 yards long,
what are the lengths of the other two sides?
Ex4: Given A = 29, a = 6, and b = 10, find B.
Ex5: Given A = 110° , c = 16 cm, and a = 25 cm, find b.
December 11, 2014
From the Law of Sines...
Area of an Oblique Triangle
A
c
h
Obtuse triangle
b
Area = 1 bc sin A = 1 ab sin C = 1 ac sin B
2
2
2
C
a
C
B
b
A
Acute Triangle
a
h
c
B
Ex6: Find the area of the triangle in which C = 143°, a = 7 meters,
and b = 18 meters.
Ex7: Two fire ranger towers lie on the east-west line and are 5 miles
apart. There is a fire with a bearing of N27E from tower 1 and N32W
from tower 2. How far is the fire from tower 1?