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THE LAW OF SINES states the following:
 The sides of a triangle are proportional to one another in the
same ratio as the sines of their opposite angles
 This means that in the oblique triangle ABC, side a, for example,
is to side b as the sine of angle A is to the sine of angle B.
 a = sin A
similarly
b = sin B
b sin B
c sin C
 You sometimes see the law of sines stated algebraically as
sin A = sin B = sin C
a
b
c
or
a
sin A
=
b
=
sin B
c .
sin C
The three angles of a triangle are 40°, 75°, and
65°. In what ratio are the three sides? Sketch the
figure and place the ratio numbers.
Solution. To find the ratios of the sides, we must
evaluate the sines of their opposite angles.
 sin 40° = .643
 sin 75° = .966
 sin 65° = .906
These are the ratios of the sides opposite those angles:
a)
B) When the side opposite the 75° angle is 10 cm, how
long is the side opposite the 40° angle?
Solution. Let us call that side x. Now, according to the
Law of Sines, in every triangle with those angles, the
sides are in the ratio 643 : 966 : 906. Therefore,
x = 643
10
966
x =6.656 cm
 The three angles of a triangle are 105°, 25°, and 50°. In
what ratio are
a) the sides? Sketch the triangle.
 b) If the side opposite 25° is 10 cm, how long is the
side opposite 50°?
When to use the law of sines formula?
 when you know 2 sides and an angle (case 1) and you want
to find the measure of an angle opposite a known side.
 when you know 2 angles and 1 side and want to get the side
opposite a known angle (case 2).
 In both cases, you must already know a side and an angle
that are opposite of each other.
Cases when you can not use the Law of Sines
The picture below illustrates a case not suited for the law
of sines. Since we do not know an opposite side and
angle, we cannot employ the law of sines formula.
Example 2. Use the Law of Sines formula to solve for b
Is it possible to use the law of sines to calculate x pictured in the
triangle below?
Answer:
Yes, first you must remember that the sum of the interior angles of
a triangle is 180 in order to calculate the measure of the angle
opposite of the side of length 19.
Now that we have the measure of that angle, use the law of sines
to find value of x
When there are no Triangles!
So if you are given any 2 sides and an angle, can you
always use the law of sines to find all the other sides
and angles of a triangle?
The answer is Not always!
Consider the following. Let's imagine that we know
that there is some 'triangle' ABC with the following
information: BC = 23 ,AC = 3, angle B = 44.
Therefore no triangle can be drawn with these
givens.
 SAS - Side, Angle, Side
ASA - Angle, Side, Angle
AAS - Angle, Angle, Side
SSS - Side, Side, Side
HL – Hypotenuse Leg for Right Triangles
 We also discovered that SSA did not work to prove
triangles congruent.
We politely called it the Donkey Theorem ; - )
 By definition, the word ambiguous means open to
two or more interpretations.
Such is the case for certain solutions when working
with the Law of Sines.
 • ASA or AAS, the Law of Sines will nicely provide
you with ONE solution for a missing side.
 SSA (where you must find an angle), the Law of Sines
could possibly provide you with one , 2 solutions or
even no solution.
The law of cosines for calculating one side of a triangle
when the angle opposite and the other two sides are
known. It can be used in conjunction with the law of
sines to find all sides and angles.
a2 = b2 + c2 – 2bc cos A
 b2 = a2 + c2 – 2ac cos B
 c2 = a2 + b2 – 2ab cos C
Cos A = b2 + c2 - a2
2bc
Cos B = a2 + c2 - b2
2ac
Cos C = a2 + b2 - c2
2ab
 Find the angles of triangle RST
s = 70 ; t = 40.5 ; r = 90
A ship R leaves port P and travels at the rate of 16 mph.
Another ship Q leaves the same port at the same time
and travels at 12mph. Both ships are traveling in
straight paths. The directions taken by the two ships
make an angle of 55 degrees. How far apart are the
ships after 3 hours?
You are heading to Beech Mountain for a ski
trip. Unfortunately, state road 105 in North Carolina is
blocked off due to a chemical spill. You have to get to
Tynecastle Highway which leads to the resort at which you
are staying. NC-105 would get you to Tynecastle Hwy in 12.8
miles. The detour begins with a 18o veer off onto a road
that runs through the local city. After 6 miles, there is
another turn that leads to Tynecastle Hwy. Assuming that
both roads on the detour are straight, how many extra
miles are you traveling to reach your destination?
 Find the remaining parts of the triangle
1.
b= 12 ; c = 27; a = 24
2. To approximate the length of a lake, a surveyor starts at
one end of the lake and walks 245 yards. He then turns
110º and walks 270 yards until he arrives at the other end of
the lake. Approximately how long is the lake?
296 yards
After the hurricane, the small tree in my neighbor’s yard was
leaning. To keep it from falling, we nailed a 6-foot strap
into the ground 4 feet from the base of the tree. We
attached the strap to the tree 3½ feet above the
ground. Determine the angle formed from the tree to the
ground.
106.1º