Download Law of Sines and Cosines Applications

16. LAW OF SINES AND COSINES
APPLICATIONS
EXAMPLE
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 CONT
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.
Law of Sines
Cross products
Divide each side by sin
Use a calculator.
Answer: The length of the shadow is about 75.9 feet.
EXAMPLE
A 5-foot fishing pole is anchored to the edge of a
dock. If the distance from the foot of the pole to the
point where the fishing line meets the water is 45 feet
and the angles given below are true, about how much
fishing line that is cast out is above the surface of the
water?
Answer: About 42 feet of the fishing line that is cast out
is above the surface of the water.
WING SPAN
C
The leading edge of
each wing of the
B-2 Stealth Bomber
A
measures 105.6 feet
in length. The angle between the wing's leading edges is 109.05°. What is the wing
span (the distance from A to C)?
5
The pitcher’s mound on a women’s softball field
is 43 feet from home plate and the distance
between the bases is 60 feet (The pitcher’s
mound is not halfway between home plate and
second base.) How far is the pitcher’s mound
from first base?
TWO SHIPS LEAVE A HARBOR AT THE SAME TIME,
TRAVELING ON COURSES THAT HAVE AN ANGLE OF
140 DEGREES BETWEEN THEM. IF THE FIRST SHIP
TRAVELS AT 26 MILES PER HOUR AND THE SECOND
SHIP TRAVELS AT 34 MILES PER HOUR, HOW FAR
APART ARE THE TWO SHIPS AFTER 3 HOURS?
Two ships leave a harbor at the same time, traveling on courses
that have an angle of 140 degrees between them. If the first ship
travels at 26 miles per hour and the second ship travels at 34 miles
per hour, how far apart are the two ships after 3 hours?
harbor
26mph*3hr = 78 miles
ship 1
140°
34mph*3hr = 102 miles
x
ship 2
Looking at the labeled picture above, we can see that the have the
lengths of two sides and the measure of the angle between them.
We are looking for the length of the third side of the triangle. In
order to find this, we will need the law of cosines. x will be side a.
Sides b and c will be 78 and 102. Angle α will be 140°.
a 2  b 2  c 2  2bc cos 
continued on next slide
Two ships leave a harbor at the same time, traveling on courses
that have an angle of 140 degrees between them. If the first ship
travels at 26 miles per hour and the second ship travels at 34 miles
per hour, how far apart are the two ships after 3 hours?
harbor
26mph*3hr = 78 miles
140°
34mph*3hr = 102 miles
x
ship 1
ship 2
x 2  782  1022  2(78)(102) cos 140 
x 2  6084  10404  15912 cos 140 
x  16488  15912 cos 140 
2
x 2  28677.29918
x   28677.29918
x  169.3437309
Since distance is positive, the
ships are approximately
169.3437309 miles apart after
3 hours.