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

Cross products

Use a calculator.

Law of Sines

Answer: The length of the shadow is about 75.9 feet.

Divide each side by sin

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.

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 The leading edge ofeach wing of theB-2 Stealth Bombermeasures 105.6 feetin length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)?

5

A

C

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?

26mph*3hr = 78 miles

harbor

ship 1 ship 2

140° 34mph*3hr = 102 miles

x

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

ship 1 ship 2

140° 34mph*3hr = 102 miles

x

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°.

harbor

26mph*3hr = 78 miles

harbor

ship 1 ship 2

140° 34mph*3hr = 102 miles

x

cos2222 bccba

26mph*3hr = 78 miles

harbor

ship 1 ship 2

140° 34mph*3hr = 102 miles

x

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

ship 1 ship 2

140° 34mph*3hr = 102 miles

x

harbor

26mph*3hr = 78 miles

harbor

ship 1 ship 2

140° 34mph*3hr = 102 miles

x

3437309.16929918.28677

29918.28677140cos1591216488

140cos15912104046084140cos)102)(78(210278

2

2

2

222

xxxxxx

Since distance is positive, the ships are approximately 169.3437309 miles apart after 3 hours.