The Law of Sines

M 140 Precalculus

V. J. Motto

What if the triangle we want to solve is NOT a right triangle? In the next two sections we’ll develop ways of solving these triangles if we know at least one side and two other pieces of info (sides or angles or one of each).

a

b

c

We’ll label these triangles with sides a, b and c and use their Greek alphabet counterparts for the angles opposite those sides as shown above.

Draw a perpendicular line and call the length h. We do this so that we have a right triangle which we already know how to work with.

h

a

b

c

Let’s write some trig functions we know from the right triangles formed.

c

hsin

a

hsinSolve these for h

hc sin ha sinSince these both = h we can substitute

sinsin ac divide both sides by ac

ac ac ca

sinsin

This process can be repeated dropping a perpendicular from a different vertex of the triangle. What we get when we combine these is:

THE LAW OF SINES

cba

sinsinsin

What this says is that you can set up the ratio of the sine of any angle in a triangle and the side opposite it and it will equal the ratio of the sine of any other angle and the side opposite it. If you know three of these pieces of information, you can then solve for the fourth.

THE LAW OF SINES

sinsinsin

cba

Use these to find missing angles

Use these to findmissing sides

There are three possible configurations that will enable us to use the Law of Sines. They are shown below.

ASA

You may have an angle, a side and then another angle

SAA

You may have a side and then an angle and then another angle

SSA

You may have two sides and then an angle not between them.

What this means is that you need to already know an angle and a side opposite it (and one other side or angle) to use the Law of Sines.

You don’t have an angle and side opposite it here but can easily find the angle opposite the side you know since the sum of the angles in a triangle must be 180°.

Solve a triangle where = 55°, = 82° and c = 9Draw a picture (just draw and label a triangle. Don't worry about having lengths and angles look right size)

a

b

c 55

82

9

Do we know an angle and side opposite it? If so the Law of Sines will help us determine the other sides. 55sin82sin

9 b 55sin982sin b

82sin

55sin9b 44.7

7.44

How can you find ?

Hint: The sum of all the angles in a triangle is 180°.

438255180

43

How can you find a? (Remember it is NOT a right triangle so Pythagorean theorem will not work).

You can use the Law of Sines again.82sin

9

43sin

a

This is SAA

82sin

43sin9a 20.6

6.20

Let's look at a triangle where you have SSA.

a

b It could be that you can't get the sides to join with the given info so these would be "no solution".

bIt could be that there is one triangle that could be formed and you could solve the triangle.

a

b

a

b a

It could be that since side c and are not given that there are two ways to draw the triangle and therefore 2 different solutions to the triangle

a, b and are the same in both of these triangles.

It is easy to remember that the tricky ones are SSA (it's bassakwards).

You can just check to see if there are two triangles whenever you have the SSA case. The "no solution" case will be obvious when computing as we will see.

Solve a triangle where = 95°, b = 4 and c = 5

95

4

5

a

We have SSA. We know an angle and a side opposite it so we'll use the Law of Sines.

5

sin

4

95sin sin

4

95sin5

245.1sin We have the answer to sine and want to know the angle so we can use inverse sine.

245.1sin 1 What happens when you put this in your calculator?

Remember the domain of the inverse sine function is numbers from -1 to 1 since the sine values range from -1 to 1. What this means is there is no solution. (You can't build a triangle like this).

a10.42

Solve a triangle where = 35°, b = 6 and c = 8

35

6

8

We have SSA again so we know it could be the weird one of no solution, one solution or two solutions.

8

sin

6

35sin sin

6

35sin8

765.0sin

Since this was an SSA triangle we need to check to see if there are two solutions. Remember your calculator only gives you one answer on the unit circle that has the sine value of 0.765. You need to figure out where the other one is and see if you can make a triangle with it.

765.0sin 1 9.49

49.9

Knowing and , can you find ? 1.959.4935180

95.1

Now how can you find a?

a

1.95sin

6

35sin 42.10

35sin

1.95sin6a

The smallest angle should have the smallest side opposite it and the largest angle should have the largest side opposite it.

Looking at the same problem: Solve a triangle where = 35°, b = 6 and c = 8

35

6

8

a

Let's check to see if there is another triangle possible. We got 49.9° from the calculator. Draw a picture and see if there is another angle whose sine is 0.765.

765.0sin

Knowing and , can you find ? 9.141.13035180

Now how can you find a?

9.14sin35sin

6 a 69.2

35sin

9.14sin6a

49.9

180 - 49.9 = 130.1

130.1

So there IS another triangle. (remember

our picture is not drawn to scale).

14.92.69

Solve a triangle where = 42°, b = 22 and c = 12

42

22

12

a

365.0sin

Knowing and , can you find ? 6.1164.2142180

12

sin

22

42sin

4.21365.0sin 1 21.4

116.629.40

6.116sin22

42sin a 40.29

42sin

6.116sin22a

Since this is SSA we need to check the other possible sine value for possibility of a second triangle solution.

Solve a triangle where = 42°, b = 22 and c = 12

42

22

12

a

Knowing and , can you find ? 6.206.15842180

This negative number tells us that there is no second triangle so this is the one triangle solution.

21.4180 - 21.4 = 158.6

158.6

Not possible to build another triangle with these stipulations.