Download 7-6 Law of Sines

The Law of Sines and fishing lines
In the remote village of Seward, Alaska there
is a five year waiting list for docking space
for fishing boats.
Law of Sines
Objectives:
• I will be able to use the Law of Sines to
solve Triangles.
• I will be able to solve problems using the
Law of Sines.
Key – The Law of Sines can be used to find missing
parts of triangles that are NOT right triangles.
The Law of Sines
The Law of Sines can be used in the following
two cases:
1. We know the measures of two angles and
any side of a triangle.
2. You know two sides and the measure of
one angle opposite one of these sides.
The Law of Sines
Let ABC be any triangle with a, b, and c being the sides
opposite the angles A, B, and C.
C
a
b
A
Then:
c
sin A sin B sin C


a
b
c
B
Depending on the type of fishing that you are doing, it is
important to determine how much fishing line you need
A 5-foot fishing pole is anchored to the edge of a dock
in Seward, Alaska. 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?
Use:
sin A sin B sin C


a
b
c
We are looking for side a.
sin 54 sin120

a
45
45(sin54) = a(sin120)
45(sin 54)
a
sin120
Answer: About 42 feet of the fishing line that is cast
out is above the surface of the water.
The Law of Sines - Alternate
Basic Form
Alternate Form:
sin A sin B sin C


a
b
c
a
b
c


sin A sin B sin C
Easier to use when trying to find a side.
A direct flight from New York to Anchorage is a shorter
than flying with a layover in Chicago. With the given
distances below, figure out how much shorter a direct flight
would be. Use the Law of Sines to figure out the distance
from Chicago to Anchorage. Round to the 10ths place.
Work with a partner, discuss and compare answers.
Anchorage
x
2°
3,360 mi.
New York
.5°
x
x
725 mi.
Chicago
Answer: About 264 miles
Example 1
Find side p. Round to the nearest tenth.
Answer:
Example 2
to the nearest degree in
Law of Sines
Cross products
Solve for L.
Answer:
,
Solving a Triangle
The Law of Sines can be used to
solve a triangle.
Solving a triangle means finding
the measures of all the sides
and all the angles of a triangle.
You may also need to use the
angle sum theorem to solve the
triangle.
This is how we solve for distances and angles
for remote mountain ranges.
Example 3
. Round
angle measures to the nearest degree and side
measures to the nearest tenth.
To
To find
find e:
d:
8
60
112
We know the measures of two angles of the triangle. Use
the Angle Sum Theorem to find
Answer:
After a tremendous earthquake struck Alaska in the mid
1960’s villages were destroyed and power lines were
broken or in the state of disrepair. The following problem
was used to determine the acceptable limits of bent and
damaged telephone poles
Example 3
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 the
Example 3
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.
Answer: The length of the shadow is about
75.9 feet.
Your Turn
Design one problem on your own
– Start by thinking what you like to do the most
• Ex. Play an instrument, sports, activity
• How/where does it form a triangle
– Next, add the dimensions for one of the following:
• The measures of two angles and any side of a triangle.
• The measures of two sides and the measure of one
angle opposite one of these sides.
– Share your sample problem with a neighbor and
see if they can solve it.
Homework
• Make 6 additional problems tonight
• 3 that provide the measures of two angles and any
side of a triangle.
• 3 that provide the measures of two sides and the
measure of one angle opposite one of these sides.
• All should relate to your own experiences…
• Look out the window, there are examples all around.