Another great rule in trigonometry is the cosine rule (also known as the Law of Cosines). This one also gives us the relationship between the angles and the sides, but we apply it in different cases from the law of sines. These cases are explained in detail in the video. The law of cosines works for any triangle, not just the right triangles. The interesting thing is that this is actually a generalization of the Pythagorean Theorem. This lecture also covers the Heron's Formula, which is used to determine the area of the triangle if you know all its sides.
This holds in any triangle - not just right triangles!
This is a generalization of the Pythagorean Theorem. (If C is a right angle, then cosC = 0, so we get c2 = a2 + b2.)
Use the Law of Cosines for triangles described in the following ways:
SAS always has a unique solution (assuming the angle is less than 180°
SSS always has a unique solution (assuming each side is less than the sum of the other two).
(Use Law of Sines for ASA, SAA, and SSA.)
where s = [1/2](a+b+c) is the semiperimeter of the triangle.
In triangle ABC, side a has length 3, side b has length 4, and angle C measures 60°
. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.
The side lengths of triangle ABC are 5, 7, and 10. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.
Find the area of a triangle whose side lengths are 5, 7, and 10.
The side lengths of triangle ABC are 16, 30, and 34. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.
A triangle has two sides of length 8 and 16 with an included angle of 45°
. Find the length of the third side and the area of the triangle.
Law of Cosines
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Side side side means you know all three sides of a triangle, but you don't necessary know any of the angles yet.0219
The point is, if you know little, little b, and little c, then you know all of these parts of the law of cosines, so you can solve for the cosine of capital C, and you can figure out what that angle capital C is.0227
Then you can figure out what one of the angles is.0245
You can just kind of rotate the triangle, and relabel what a, b, and c are to find the other two angles.0248
If you know all three sides of a triangle, the law of cosines is very useful for finding the angles, one at a time.0254
Remember, there's a couple other ways that you can be given information for triangles.0261
You can be given, angle side angle, or side angle angle, or side side angle.0266
Those two don't really lend themselves very well to solution by the law of cosines.0273
If you're given one of those situations then you really want to use the law of sines which we learned about in the previous lecture.0280
There's one more formula we're going to be using in this lecture which is Heron's formula.0289
The point here is that, if you know all three lengths of sides of a triangle, I'll call them a, b, and c, as usual, then you have a nice formula for the area.0294
Now, I've solved for all three sides of the triangle, and all three angles of the triangle.0837
It's nice at this point, even though we're done with the problem to get some kind of check because we've done lots of calculations here, we could have made a mistake.0841
What I'm going to do is add up all three angles in the triangle, and make sure that they come up to be 180 degrees.0848
To check on my work here, I'll add up 60+46+74, that does indeed come out to be 180 degrees.0856
That suggests that we probably didn't make a mistake in solving all those angles.0877
Just to recap this problem here, we're given a side angle side situation, that's a definite tip-off that you're going to be using the law of cosines.0882
I filled in my side, the included angle, and a side.0891
The first thing I did was I used the law of cosines to find the missing side.0895
To solve the triangle completely, I still had two angles that I didn't know, I used the law of sines after that to find the two angles that I didn't know based on knowing the other sides and the other side, and angle.0903
If you know three side lengths of a triangle, then what you do is you work out the semi-perimeter, you just drop the side lengths into this formula for the semi-perimeter, then you drop the semi-perimeter and the three side lengths for the Heron's formula for the area.1703
It simplifies down pretty quickly to give you the area of the triangle.1720