The Law of Cosines - Revision history

Khanh: /* Licensing */

2021-10-28T23:00:11Z

Licensing

Khanh at 22:59, 28 October 2021

2021-10-28T22:59:22Z

Khanh: /* Law of Cosines */

2021-10-28T00:40:01Z

Law of Cosines

Lila at 23:09, 7 October 2021

2021-10-07T23:09:18Z

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T22:25:47Z

fixing link to go straight to pdf

Rylee.taylor: added The Law of Cosines notes by professor Esparza

2020-08-11T21:59:26Z

added The Law of Cosines notes by professor Esparza

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/ The Law of Cosines]. Written notes created by Professor Esparza, UTSA.

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 == Licensing ==
 Content obtained and/or adapted from:
-* [* [https://en.wikibooks.org/wiki/Trigonometry/Law_of_Cosines Law of Cosines, Wikibooks] under a CC BY-SA license
+* [https://en.wikibooks.org/wiki/Trigonometry/Law_of_Cosines Law of Cosines, Wikibooks] under a CC BY-SA license

@@ Line 57: / Line 57: @@
 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3A.pdf The Law of Cosines]. Written notes created by Professor Esparza, UTSA.
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3B.pdf The Law of Cosines Continued]. Written notes created by Professor Esparza, UTSA.

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 [[Image:Law-of-cosines1.svg]]
-The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:<ref name=Leff1>
+The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:
 :<math>a^2+b^2-2ab\cos(\theta)=c^2</math>
 where <math>\theta</math> is the angle between sides <math>a</math> and <math>b</math> .

← Older revision		Revision as of 23:09, 7 October 2021
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		+	==Law of Cosines==
		+	[[Image:Law-of-cosines1.svg]]
		+
		+	The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:<ref name=Leff1>
		+
		+	{{cite book \|title=''cited work'' \|author=Lawrence S. Leff \|url=http://books.google.com/?id=y_7yrqrHTb4C&pg=PA326 \|page=326 \|isbn=0764128922 \|publisher=Barron's Educational Series \|date=2005-05-01}}
		+	</ref>
		+	:<math>a^2+b^2-2ab\cos(\theta)=c^2</math>
		+	where <math>\theta</math> is the angle between sides <math>a</math> and <math>b</math> .
		+
		+	===Does the formula make sense?===
		+	This formula had better agree with the Pythagorean Theorem when <math>\theta=90^\circ</math> .
		+
		+	So try it...
		+
		+	When <math>\theta=90^\circ</math> , <math>\cos(\theta)=\cos(90^\circ)=0</math>
		+
		+	The <math>-2ab\cos(\theta)=0</math> and the formula reduces to the usual Pythagorean theorem.
		+
		+	==Permutations==
		+	For any triangle with angles <math>A,B,C</math> and corresponding opposite side lengths <math>a,b,c</math> , the Law of Cosines states that
		+	:<math>a^2=b^2+c^2-2bc\cdot\cos(A)</math>
		+	:<math>b^2=a^2+c^2-2ac\cdot\cos(B)</math>
		+	:<math>c^2=a^2+b^2-2ab\cdot\cos(C)</math>
		+
		+	===Proof===
		+	[[Image:Law-of-cosines2.svg]]
		+
		+	Dropping a perpendicular <math>OC</math> from vertex <math>C</math> to intersect <math>AB</math> (or <math>AB</math> extended) at <math>O</math> splits this triangle into two right-angled triangles <math>AOC</math> and <math>BOC</math> , with altitude <math>h</math> from side <math>c</math> .
		+
		+	First we will find the lengths of the other two sides of triangle <math>AOC</math> in terms of known quantities, using triangle <math>BOC</math> .
		+
		+	:<math>h=a\sin(B)</math>
		+	Side <math>c</math> is split into two segments, with total length <math>c</math> .
		+	:<math>\overline{OB}</math> has length <math>\overline{BC}\cos(B)=a\cos(B)</math>
		+	:<math>\overline{AO}=\overline{AB}-\overline{OB}</math> has length <math>c-a\cos(B)</math>
		+
		+	Now we can use the Pythagorean Theorem to find <math>b</math> , since <math>b^2=\overline{AO}^2+h^2</math> .
		+	:{\|
		+	\|<math>b^2</math>
		+	\|<math>=\bigl(c-a\cos(B)\bigr)^2+a^2\sin^2(B)</math>
		+	\|-
		+	\|
		+	\|<math>=c^2-2ac\cos(B)+a^2\cos^2(B)+a^2\sin^2(B)</math>
		+	\|-
		+	\|
		+	\|<math>=a^2+c^2-2ac\cos(B)</math>
		+	\|}
		+
		+	The corresponding expressions for <math>a</math> and <math>c</math> can be proved similarly.
		+
		+	The formula can be rearranged:
		+	:<math>\cos(C)=\frac{a^2+b^2-c^2}{2ab}</math>
		+	and similarly for <math>cos(A)</math> and <math>cos(B)</math> .
		+
		+	==Applications==
		+	This formula can be used to find the third side of a triangle if the other two sides and the angle between them are known. The rearranged formula can be used to find the angles of a triangle if all three sides are known.
		+
		+
		+	==Resources==
		+	* [https://en.wikibooks.org/wiki/Trigonometry/Law_of_Cosines Law of Cosines], WikiBooks: Trigonometry
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3A.pdf The Law of Cosines]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3A.pdf The Law of Cosines]. Written notes created by Professor Esparza, UTSA.
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3B.pdf The Law of Cosines Continued]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3B.pdf The Law of Cosines Continued]. Written notes created by Professor Esparza, UTSA.

← Older revision		Revision as of 22:25, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/ The Law of Cosines]. Written notes created by Professor Esparza, UTSA.	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3A.pdf The Law of Cosines]. Written notes created by Professor Esparza, UTSA.
		+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Cosines/Esparza%201093%20Notes%204.3B.pdf The Law of Cosines Continued]. Written notes created by Professor Esparza, UTSA.