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Pre-Calculus: The Law of Cosines


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:39
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 61 MB
  • Posted: 01/22/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Pre-Calculus Review (31 lessons, $61.38)
Trigonometry: Applications of Trigonometry (14 lessons, $26.73)
Trigonometry: The Law of Cosines (4 lessons, $5.94)

In this lesson, Professor Burger begins with a review of the Law of Sines. He then introduces the Law of Cosines, which extends the Pythagorean Theorem to triangles that are not right, allowing us to solve for any angle in the triangle. The Law of Cosines, also called the Al-Kashi Law and the Cosine Formula and the Cosine Rule, states that, for the angle you are solving for, opposite side ^2 = (sum of the squares of adjacent sides) - 2 * (product of the adjacent sides) (cos of the desired angle). The law of cosines is most useful when computing the third side of a triangle when two sides and their enclosed angle are known (SAS) or when computing the angles of a triangle in which all sides are known (SSS).

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Recent Reviews

Need more practice problems
~ tansley

The information is fantastic. However - practice makes perfect - so it would be nice to see a few examples.

Need more practice problems
~ tansley

The information is fantastic. However - practice makes perfect - so it would be nice to see a few examples.

Applications of Trigonometry
The Law of Cosines
The Law of Cosines Page [1 of 2]
Okay, so I want to remind you about the Law of Sines. That just tells you, if you have any triangle at all and you know
the angles and the lengths of the opposite sides, you always have these ratios equal. Sine of ? divided by the length
of the side opposite ? is going to equal the sine of ? divided by the length of the side opposite ? , and that will
equal the sine of ? divided by the length of the side opposite ? . And you can use that, in fact, to do all sorts of crazy
things, and we talked about those.
Now, this does raise the question: If there exists a Law of Sines, does there exist a Law of Cosines? And if such a
law were to be put on the books, would it be sort of an awful law? Like, you know, you’d have to—I don’t know what—
kill small animals or something? Well, it turns out that this is where cosine sort of redeems itself. It’s been getting bad
press lately here in the math community, as you know, in terms of its sum of angles and so forth and so on. But it
turns out that the Law of Cosines is a wonderful thing. And in fact what the Law of Cosines allows us to do is in fact
fulfill a fantasy that you might have felt sort of brewing inside of you, sort of feeling this urge inside of you. You want
it, you want it, you want it! But you didn’t know how to get it. And that is, how do you extend the Pythagorean
theorem to triangles that aren’t right?
Now, the Pythagorean theorem is probably one of the coolest theorems around. It just says that if you have a, b and
c—those are the lengths of a right triangle—then a2+b2=c2. And I bet you have fantasized more than once—you
can tell me; we’re friends—that in fact, gosh, if only any arbitrary triangle would satisfy this kind of thing, wouldn’t
“blank” be a better place? Well, it turns out that ordinary triangles do satisfy a similar thing, and in fact it’s sort of a
generalization of the Pythagorean theorem, and this is known as the Law of Cosines. Let me actually show you the
generalization of the Pythagorean theorem in this setting.
Suppose you have now any old triangle, by no means right—this could be a wrong triangle! This is a wrong joke!
Gems! All right, now suppose that I call this angle ? . I’ll call this angle ? and I’ll call this angle ? . So this has
length a, this has length b, and this has length c. Well, then, the Law of Cosines actually can be given in three
different formulations. What I’m basically going to do is I’m going to extend this. See, here I had a very special angle
of 90°. Here none of the angles are particularly special. So there’s sort of three versions, depending upon which
angle you’re going to think is special.
Let me first tell you how they look. One way is to think of this angle as being sort of a 90° special angle. If you think
of that as the special angle, then the Law of Cosines tells you the following relationship between the length of the
sides and the angles: a2=b2+c2. That is the fantasy you’ve always wanted. But there’s a small price you pay. I
have to subtract off 2bccos? . So when I’m looking at the Law of Cosines from the vantage point of ? , what I see is
that the opposite side, that squared, will equal the sum of the squares of the other sides, minus 2 times the product of
the other sides multiplied by the cosine of that angle.
So now what would it look like if I picked, let’s say, ? as my vantage point? Then I would say
b2=a2+c2?2accos? . And finally, if I think of the ? angle as sort of the biggie angle, then I’d see c2=a2+b2—
that is really exactly the Pythagorean Theorem, but ?2abcos? . So it’s a thing that you can sort of remember. If you
know the angle that you want to expand around, it’s the opposite side squared will equal the sum of the square of the
adjacent sides, minus 2 times the lengths of the adjacent sides times the cosine of that angle. And that always holds,
no matter which angle you pick.
And you can see, if you have a right triangle—well, let’s use this expansion here, take this as the angle. Then I have
cos90! . And what’s cos90! ? Well, hopefully you remember that, but if not, take a look at the graph. cos90! is
zero. So then that’s zero. So in this very special case when I go around this angle, this term just drops out, and then
I’m left with the Pythagorean theorem. So you can see this really generalizes the Pythagorean theorem but now it’s to
arbitrary triangles. So it’s really, really powerful. It allows you to get relationships between lengths of sides and
measures of angles in this fashion.
Applications of Trigonometry
The Law of Cosines
The Law of Cosines Page [2 of 2]
Up next we’ll start taking a look at a lot of different applications and varieties of how to use the Law of Cosines in order
to find some lengths and some measures just knowing a couple of others. I’ll see you there.

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