1. Law of Sines
Section 3.1

2. Deriving the Law of SinesC b a h α β c B A
Since we could draw another altitude and perform the same operations, we can extend the last equality by including the other ratio. This gives us the LAW OF SINES

3. Example 1
Find the missing parts of triangle ABC if
Rounding to the nearest integer c = 52 in

4. If you use a chart or table it looks like this:

5. Example 2
Solve the following triangle:
Since we know two of the angles of the triangle, we can find the third angle by subtracting the two known angles from 180.
Now we can use the Law of Sines to solve for either a or c. We will need to use the Law of Sines twice, once for each remaining side.

6. Use your calculator to solve for a and c.
Since the given side is an integer, round each value to the nearest integer.

7. Example 3
Solve the following triangle:
Since we know both c and γ we can use the law of sines.
We find β by using
The problem we have is that there is a first quadrant angle and a second quadrant angle with this sine.

8. Find both angles that have the sine we found.
Is it possible for both values to work in this triangle?
We check by finding the sum of β and each of the values we found. If the sum is less than 180, then the angle can be part of this triangle.
Since each of these is less than 180, there is a third angle that will make a triangle in both cases. Find each angle and then use the Law of Sines to find the remaining side in each case.

10. Example 4
We have done two examples with one angle and two sides given with different results. The first time there was one triangle and the second time two triangles. Let’s look at a third case.
Since we know both b and β, we can use the Law of Sines to find α.
At this point you can use the sin-1 function on your calculator or you can find sinα.

11. sin α =
Since the sine is never greater than 1, we know there is a problem. This means there can be no triangle with the given information.
If you use the sin-1 method, your calculator gives you an error message. This also means there can be no triangle.
This case is called the ambiguous case because there are three possible outcomes. When given two sides and an angle opposite one of the sides, there could be none, one, or two triangles. You must always check to see which case you have.

12. Example 5
A hot air balloon is sighted at the same time by two friends who are 2 miles apart on the same side of the balloon. The angles of elevation from the two friends to the balloon are 20.5° and 25.5° respectively. How high is the balloon?
There are several triangles in the diagram, including two right triangles, but none of them have enough given information that can be used to solve for anything. If you look at the small triangle on the left, you should be able to observe that we can find the remaining two angles. Then we can solve for a side. h
20.5°
25.5°
2 mi

13. With this information, we can solve for one of the two unknown sides of the small triangle. The problem can be solved using either side, so it doesn’t matter which one we pick. Let’s choose the one on the left of the diagram. We’ll call it x.
Using the right triangle with x the hypotenuse and h the side opposite the 20.5° angle, we can find h.
The balloon is 3.5 mi high.

14. You have some problems assigned that you should now work
You have some problems assigned that you should now work. Practice will help you understand the Law of Sines. Good Luck!