Difference between revisions of "The Law of Cosines"

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==Law of Cosines==
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[[Image:Law-of-cosines1.svg]]
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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>
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{{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}}
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</ref>
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:<math>a^2+b^2-2ab\cos(\theta)=c^2</math>
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where <math>\theta</math> is the angle between sides <math>a</math> and <math>b</math> .
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===Does the formula make sense?===
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This formula had better agree with the Pythagorean Theorem when <math>\theta=90^\circ</math> . 
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So try it...
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When <math>\theta=90^\circ</math> , <math>\cos(\theta)=\cos(90^\circ)=0</math>
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The <math>-2ab\cos(\theta)=0</math> and the formula reduces to the usual Pythagorean theorem.
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==Permutations==
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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
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:<math>a^2=b^2+c^2-2bc\cdot\cos(A)</math>
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:<math>b^2=a^2+c^2-2ac\cdot\cos(B)</math>
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:<math>c^2=a^2+b^2-2ab\cdot\cos(C)</math>
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===Proof===
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[[Image:Law-of-cosines2.svg]]
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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> .
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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> .
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:<math>h=a\sin(B)</math>
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Side <math>c</math> is split into two segments, with total length <math>c</math> .
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:<math>\overline{OB}</math> has length <math>\overline{BC}\cos(B)=a\cos(B)</math>
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:<math>\overline{AO}=\overline{AB}-\overline{OB}</math> has length <math>c-a\cos(B)</math>
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Now we can use the Pythagorean Theorem to find <math>b</math> , since <math>b^2=\overline{AO}^2+h^2</math> .
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:{|
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|<math>b^2</math>
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|<math>=\bigl(c-a\cos(B)\bigr)^2+a^2\sin^2(B)</math>
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|-
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|
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|<math>=c^2-2ac\cos(B)+a^2\cos^2(B)+a^2\sin^2(B)</math>
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|-
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|
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|<math>=a^2+c^2-2ac\cos(B)</math>
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|}
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The corresponding expressions for <math>a</math> and <math>c</math> can be proved similarly.
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The formula can be rearranged:
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:<math>\cos(C)=\frac{a^2+b^2-c^2}{2ab}</math>
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and similarly for <math>cos(A)</math> and <math>cos(B)</math> .
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==Applications==
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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.
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==Resources==
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* [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.

Revision as of 17:09, 7 October 2021

Law of Cosines

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:[1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2+b^2-2ab\cos(\theta)=c^2}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is the angle between sides Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} .

Does the formula make sense?

This formula had better agree with the Pythagorean Theorem when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=90^\circ} .

So try it...

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=90^\circ} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta)=\cos(90^\circ)=0}

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2ab\cos(\theta)=0} and the formula reduces to the usual Pythagorean theorem.

Permutations

For any triangle with angles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,B,C} and corresponding opposite side lengths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c} , the Law of Cosines states that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2=b^2+c^2-2bc\cdot\cos(A)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2=a^2+c^2-2ac\cdot\cos(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^2=a^2+b^2-2ab\cdot\cos(C)}

Proof

Law-of-cosines2.svg

Dropping a perpendicular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle OC} from vertex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} to intersect Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB} (or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB} extended) at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} splits this triangle into two right-angled triangles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AOC} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BOC} , with altitude Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} from side Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

First we will find the lengths of the other two sides of triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AOC} in terms of known quantities, using triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BOC} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h=a\sin(B)}

Side Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is split into two segments, with total length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{OB}} has length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{BC}\cos(B)=a\cos(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AO}=\overline{AB}-\overline{OB}} has length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c-a\cos(B)}

Now we can use the Pythagorean Theorem to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2=\overline{AO}^2+h^2} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\bigl(c-a\cos(B)\bigr)^2+a^2\sin^2(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =c^2-2ac\cos(B)+a^2\cos^2(B)+a^2\sin^2(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =a^2+c^2-2ac\cos(B)}

The corresponding expressions for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} can be proved similarly.

The formula can be rearranged:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(C)=\frac{a^2+b^2-c^2}{2ab}}

and similarly for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cos(A)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cos(B)} .

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

  • ↑ Template:Cite book
    • Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=The_Law_of_Cosines&oldid=1955"