Difference between revisions of "Law of Cosines"
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Because of the identity <math>|\vec{a}|^2=\vec{a}\cdot\vec{a}</math>,we can complete our proof as the following. | Because of the identity <math>|\vec{a}|^2=\vec{a}\cdot\vec{a}</math>,we can complete our proof as the following. | ||
− | Draw the diagram. Note that <math>\vec{c}+\vec{a}=\vec{b}</math>. Then <math>\vec{b}-\vec{c}=\vec{a}</math>. <math>\vec{a} | + | Draw the diagram. Note that <math>\vec{c}+\vec{a}=\vec{b}</math>. Then <math>\vec{b}-\vec{c}=\vec{a}</math>. <math>\vec{a}\cdot \vec{a}=a^2</math>. <math>(\vec{b}-\vec{c})^2=b^2+c^2-2\cdot b\cdot c\cdot \cos{A}=|\vec{a}|^2</math>. Now, we have finished the proof because the two quantities are equal. |
Credits to China High School Math textbook <math>\emph{Mathematics Vol 5B Textbook}</math> by People's Education Press (this textbook is currently discontinued but it has been used for hinting the proof. The proof is done by myself. But the letting and the process of guiding students to verify the identity <math>|\vec{a}|^2=\vec{a}\cdot\vec{a}</math> is written in the textbook. | Credits to China High School Math textbook <math>\emph{Mathematics Vol 5B Textbook}</math> by People's Education Press (this textbook is currently discontinued but it has been used for hinting the proof. The proof is done by myself. But the letting and the process of guiding students to verify the identity <math>|\vec{a}|^2=\vec{a}\cdot\vec{a}</math> is written in the textbook. |
Revision as of 09:05, 7 June 2023
The Law of Cosines is a theorem which relates the side-lengths and angles of a triangle. It can be derived in several different ways, the most common of which are listed in the "proofs" section below. It can be used to derive the third side given two sides and the included angle. All triangles with two sides and an include angle are congruent by the Side-Angle-Side congruence postulate.
Contents
Theorem
For a triangle with edges of length ,
and
opposite angles of measure
,
and
, respectively, the Law of Cosines states:
In the case that one of the angles has measure (is a right angle), the corresponding statement reduces to the Pythagorean Theorem.
Proofs
Proof 1
Acute Triangle
Let ,
, and
be the side lengths,
is the angle measure opposite side
,
is the distance from angle
to side
, and
and
are the lengths that
is split into by
.
We use the Pythagorean theorem:
We are trying to get on the LHS, because then the RHS would be
.
We use the addition rule for cosines and get:
We multiply by -2ab and get:
Now remember our equation?
We replace the by
and get:
We can use the same argument on the other sides.
Right Triangle
Since ,
, so the expression reduces to the Pythagorean Theorem. You can find several proofs of the Pythagorean Theorem here.
Obtuse Triangle
The argument for an obtuse triangle is the same as the proof for an acute triangle.
Proof 2 (Vector Dot Product)
Consider . Let
.
Because of the identity ,we can complete our proof as the following.
Draw the diagram. Note that . Then
.
.
. Now, we have finished the proof because the two quantities are equal.
Credits to China High School Math textbook by People's Education Press (this textbook is currently discontinued but it has been used for hinting the proof. The proof is done by myself. But the letting and the process of guiding students to verify the identity
is written in the textbook.
~hastapasta
Problems
Introductory
1. If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?
![$\mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } 8/\sqrt{15} \qquad \mathrm{(C) \ } 5/2 \qquad \mathrm{(D) \ } \sqrt{6} \qquad \mathrm{(E) \ } (\sqrt{6} + 1)/2$](http://latex.artofproblemsolving.com/a/d/3/ad39e455fe156ef0c687ce7c7d910ebd4f9c9f27.png)
(Source)
2. In the quadrilateral ,
,
,
.
(1) Find .
(2) If , find
.
(2018 China Gaokao Syllabus I #17)
Solution link: https://artofproblemsolving.com/community/c4h2812553_trig__triangle_laws_how_come P.S.: Since the solution is on a forum, please read all the way to thread #3 for the solutions!
Intermediate
A tripod has three legs each of length feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is
feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let
be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then
can be written in the form
where
and
are positive integers and
is not divisible by the square of any prime. Find
(The notation
denotes the greatest integer that is less than or equal to
)
(Source)
Olympiad
A tetrahedron is inscribed in the sphere
. Find the locus of points
, situated in
, such that
where are the other intersection points of
with
.
(Source)