Download 3-7-16 NOTES law of sines 2 column final

Unit 6 – Law of Sines &
Precalculus
Cosines
Name
Date
Monday March 7, 2016
Notes Law of Sines & Area of an Oblique Triangle
The Law of Cosines can be used if you are given:
The Law of Sines can be used if you are given:


 Two angles and a non-included side (AAS)
 Two angles and an included side (ASA)
 Two sides and a non-included angle (SSA)*
(SSA is referred to as the ambiguous case since it could have no
Three sides (SSS)
Two sides and an included angle (SAS)
triangle, one triangle, or two triangles. This is why it couldn’t be
used to prove triangles congruent back in Geometry.)
or
Important Note: You MUST use the Law of Cosines to find the largest
angle of the triangle BEFORE switching to Law of Sines. Failing to
follow this rule could result in creating an unnecessary and incorrect
ambiguous case accidentally. 
Note: You will only use two of the three ratios at a time.
If given info is ASA or AAS, then start with law of sines and know that there is only one triangle possible.
But if given info is SSA, this is the ambiguous case, and you need to test to see how many (if any) triangle(s) exist.
1. Solve each triangle. If multiple solutions exist, find them.
Round any side lengths to the nearest hundredth and round angle measurements to the nearest tenth of a degree.
Appropriate work must be shown to receive full credit for correct responses.
Start with
# of ∆(s):
Since given info is AAS (Angle, Angle, Non-included Side), start with law of sines
AAS will always have 1 triangle.
Since both “c” and angle C are unknown, find side “a” first.
a
b

sin A sin B
Law of Cosines
no triangle
1
2
0
= A =
43
= B =
1090
= C =
=a=
=b=
15
=c=
Use this version where the “sides” are in the numerator since that is what we’re finding first.
a
15


sin 43 sin109

15sin 43
sin109
a  10.82 ft
a
Law of Sines
Recall that the sum of all the angles in a triangle is 1800
Always continue to use the two “known” corresponding pieces when finding
missing info. Minimize using a value you found since it is possible you
made a mistake that lead to additional errors if used.
b
c

sin B sin C
C  180  109  43

C  28
15
c


sin109 sin 28
15sin 28
c
sin109
c  7.45 ft
If you have an ambiguous case, start with Law of Sines normally.
Case 1: If you receive an error when solving for an angle it indicates that “no triangle” exists.
Case 2: Once you find the first angle, record this in the chart.
Then find its supplement (1800 – angle) and put it as a possible angle for the 2nd triangle.
Find the 3rd angle of each triangle. (If the sum of the two angles is one triangle is already over 180 0,
then you actually only have one triangle and can cross out the second column.)
Proceed with finding any remaining sides of the triangle(s).
2.
a = 15, b = 25, and A = 85 
Start with
# of ∆(s):
Law of Sines
Law of Cosines
no triangle
1
2
0
= A =
85
= B =
= C =
=a=
15
=b=
25
=c=
3.
a = 12, b = 31, and A = 20.5 
Start with
# of ∆(s):
Law of Sines
Law of Cosines
no triangle
1
= A =
2
20.50
= B =
= C =
=a=
12
=b=
31
=c=
B
a
c
h
Consider the triangle ABC given at the right.
What is the formula you learned years ago to find the area for this figure?
Using right triangle trigonometry,
what are two different ways we could write an expression for the height of this triangle?
or
Incorporate this “new” height into your previous Geometry formula for finding the area of a triangle.
Voila! You have just discovered the area formula you can use when the height of a triangle is not given to you. 
Area of an Oblique Triangle:
The area of any triangle is given by
one-half the product of the lengths of two sides times the sine of their included angle.
1
1
1
Area  ab sin C or Area  bc sin A or Area  ac sin B
2
2
2
Find the area of each oblique triangle.
4.
(You may need to use Law of Sines first to find a missing piece.)
a = 90 m, b = 52 m, C = 1020
Round to the nearest square unit.
5.
B = 870, C = 290, a = 34 cm
Round to the nearest hundredth.