The Trigonometric Functions
The Law of Sines
Where do we use the Law of Sines
„
The Law of Sines can be used to solve
triangles that are not right triangles.
The Formula
Let triangle ABC be any triangle with a, b, c
representing the measures of the sides opposite
the angles with measures A,B,C, respectively.
Then the following is true.
a
b
c
=
=
sin a sinb sin c
Example #1
„
Solve ∆LMN if L= 29 degrees, M= 112 degrees and
l=22 . Since a the sum of the angles of a triangle is
180 degrees, we know that angle n= 39 degrees.
„
M
112
29
L
22
m
=
sin 29 sin 112
m ≈ 42.1
22
N
22
n
=
o
o
sin 29
sin 39
n ≈ 28.6
Area of Triangles
Let triangle ABC be any triangle with a, b, c
representing the measures of the sides opposite the
angles with measurements A,B,C, respectively. Then
the area K can be determined using one of the
1
following formulas.
K = bc sin A
2
1
K = ab sin C
2
1
„ Note this is SAS
K = ac sin B
2
situation.
„
Example #2
„
Find the area of triangle ABC if b=21.2, c= 16.5
and angle A = 25 degrees.
„
K=½ (21.2)(16.5) SIN25
K≈ 729 UNITS SQUARED
„
Area of Triangles
„
Use this formula if you know the measure of only one
side and two angles. The ASA or AAS situation.
1 2 sin B sin C
K= a
2
sin A
1 2 sin A sin C
K= b
2
sin B
1 2 sin A sin B
K= c
2
sin C
Example #3
„
„
Find the area of triangle JKL if j=45, K= 111.1
degrees, and L= 27.3 degrees
2
K=1/2(45.7)
SIN
111.1 SIN 27.3
SIN 41.6
Real Life Application
„
A person in a hot air balloon observes that the
angle of depression to a building on the ground
x
tan 24 .2 o =
is 65.8 degrees. After ascending vertically 500
y
feet, the person now observes that the angel of
x
.44942 =
depression is 70.2 degrees. How far is the
y
balloonist now from the building? Find D
x = .44942 y
x
500 + y
x
.36002 =
500 + y
.44942 y
.36002 =
500 + y
180 .01 + .36002 y = .44942 y
180 .01 = .08940 y
y = 2013 .53
tan 19 .8 o =
„
500
19.8
70.2
D
65.8
24.2
y
x
Classroom building
Continuation
„
„
Now we can substitute in the y and find x which is
904.97
If we look at our picture, we have side 500+y and
side x and we need the distance from the hot air
balloon to the classroom. Using the Pythagorean
theorem we have
(2513.6) 2 + (904.97) 2 = dis tan ce 2
d 2 = 6318184.96 + 818970.70 = 7137155.66
d ≈ 2671.55
HW# 37
Section 5.6
„ Pgs. 316-318
„ #11-21 odd, 27,29,32,38
„