Two great law of sines problems

These two law of sines problems below will show you how to use the law of sines to solve some real life problems.

Problem #1

Two fire-lookout stations are 15 miles apart, with station A directly east of station B. Both stations spot a fire. The angular direction of the fire from station B is N52°E and the angular direction of the fire from station A is N36°W. How far is the fire from station A? 


The biggest trick is this problem is to understand the meaning of  N52°E and N36°W.

N52°E means 52 degrees east of north.

N36°W means 36 degrees west of north.

Here is what the graph will look like.

Law of sines problem

Notice that to find 38°, you need to subtract 52° from 90°. By the same token, to find 54°, you need to subtract 36° from 90°.

Notice also that the angle opposite 15 is missing, so we need to find it.

38 + 54 + missing angle = 180

92 + missing angle = 180, so missing angle  = 88

Now, we can use the law of sines to find the distance the fire is from station A.

Let x be the distance from the fire to station A.

sin(88°) / 15
= 0.066

sin(38°) / X
= 0.066

x =
sin(38°) / 0.066
= 0.066

The distance the fire is from station A is = 9.328 miles

Problem #2

The leaning tower of pisa is inclined 5.5 degrees from the vertical. At a distance of 100 meters from the wall of the tower, the angle of elevation to the top is 30.5 degrees. Use the law of sines to estimate the height of the leaning tower? 


The trickiest thing here is making the graph. We show it below. Notice that the height is shown with a green line.

Law of sines and leaning tower of pisa

Let us find angle y and angle z

angle y + 5.5 = 90, so angle y = 84.5

84.5 + 30.5 + angle z =180

115 + angle z = 180

angle z =  65

Now, we can use law of sines.

sin(65°) / 100
= 0.00906

sin(30.5°) / height of tower
= 0.00906

Height of tower =
sin(30.5°) / 0.00906

Height of tower = 56 meters

In reality, the height of the leaning tower of pisa is 55.86 meters

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