1. Laws of Sines and Cosines

2. Formulas Law of sines Law of cosines Heron’s formula or or
a, b, and c are the lengths of the sides of the triangle
P is the perimeter of the triangle
A is the area of a triangle

3. Use the Law of Cosines to find the value of the side x.
Now we plug into the law of cosine formula to find x.
Since length is positive, x is approximately

4. 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?
For this problem, the first thing that we should do is draw a picture. Once we have the picture, we may be able to see which formula we can use to solve the problem.
continued on next slide

5. 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
26mph*3hr = 78 miles
harbor
ship 1
ship 2
140°
34mph*3hr = 102 miles x
harbor
harbor
140°
34mph*3hr = 102 miles x
ship 1
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 Angle α will be 140°.
continued on next slide

6. 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
26mph*3hr = 78 miles
harbor
ship 1
ship 2
140°
34mph*3hr = 102 miles x
harbor
harbor
140°
34mph*3hr = 102 miles x
ship 1
ship 2
Since distance is positive, the ships are approximately miles apart after 3 hours.

7. Approximating the area of a triangle
Heron’s Formula
where P is the perimeter of the triangle and a,b, and c are the lengths
of the sides of the triangle. OR
The area of a triangle equals one-half the product of the lengths of any two sides and the sine of the angle between them.
i.e. 1/2ab sinγ = A