Difference between revisions of "The Law of Sines"

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[[Image:Law-of-sines1.svg]]
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For any triangle with vertices <math>A,B,C</math> corresponding angles <math>A,B,C</math> and corresponding opposite side lengths <math>a,b,c</math> , the Law of Sines states that
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:<math>\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}</math>
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Each of these expressions is also equal to the diameter of the triangle's [[Trigonometry/Circles and Triangles/The Circumcircle|circumcircle]] (the circle that passes through the points <math>A,B,C</math>). The law can also be written in terms of the reciprocals:
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:<math>\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}</math>
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==Proof==
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[[Image:Law-of-sines2.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> .
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We can calculate the length <math>h</math> of the altitude <math>OC</math> in two different ways:
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*Using the triangle AOC gives
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:<math>h=b\sin(A)</math> ;
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*and using the triangle BOC gives
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:<math>h=a\sin(B)</math> .
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*Eliminate <math>h</math> from these two equations:
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:<math>a\sin(B)=b\sin(A)</math> .
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*Rearrange to obtain
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:<math>\frac{a}{\sin(A)}=\frac{b}{\sin(B)}</math>
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By using the other two perpendiculars the full law of sines can be proved. '''QED.'''
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==Application==
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This formula can be used to find the other two sides of a triangle when one side and the three angles are known. (If two angles are known, the third is easily found since the sum of the angles is <math>180^\circ</math> .) See [[Trigonometry/Solving Triangles Given ASA|Solving Triangles Given ASA]]. It can also be used to find an angle when two sides and the angle opposite one side are known.
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==Area of a triangle==
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The area of a triangle may be found in various ways. If all three sides are known, use Heron's theorem.
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If two sides and the included angle are known, consider the second diagram above. Let the sides <math>b</math> and <math>c</math> , and the angle between them <math>\alpha</math> be known. The terms /alpha and /gamma are variables represented by Greek alphabet letters, and these are commonly used interchangeably in trigonometry just like English variables x, y, z, a, b, c, etc. From triangle <math>ACO</math> , the altitude <math>h=CO</math> is <math>b\sin(\alpha)</math> so the area is <math>\frac{bc\sin(\alpha)}{2}</math> .
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If two angles and the included side are known, again consider the second diagram above. Let the side <math>c</math> and the angles <math>\alpha</math> and <math>\gamma</math> be known. Let <math>AO=x</math> . Then
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:<math>\frac{x}{h}=\tan(\alpha)\text{ ; }\frac{c-x}{h}=\tan(\gamma)\text{ ; adding these, }\frac{c}{h}=\tan(\alpha)+\tan(\gamma)</math>
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Thus
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:<math>h=\frac{c}{\tan(\alpha)+\tan(\gamma)}\text{ so  area }=\frac{c^2}{2\bigl(\tan(\alpha)+\tan(\gamma)\bigr)}</math> .
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==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Sines/Esparza%201093%20Notes%204.2.pdf The Law of Sines]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Sines/Esparza%201093%20Notes%204.2.pdf The Law of Sines]. Written notes created by Professor Esparza, UTSA.
 
* [https://www.youtube.com/watch?v=1WXhKIK8oEM Application of the Law of Sines]. Produced by Professor Zachary Sharon, UTSA.
 
* [https://www.youtube.com/watch?v=1WXhKIK8oEM Application of the Law of Sines]. Produced by Professor Zachary Sharon, UTSA.

Revision as of 17:04, 7 October 2021

Law-of-sines1.svg

For any triangle with vertices corresponding 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 Sines 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 \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}}

Each of these expressions is also equal to the diameter of the triangle's circumcircle (the circle that passes through the points 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 can also be written in terms of the reciprocals:

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 \frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}}

Proof

Law-of-sines2.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 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 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} . We can calculate the 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 h} of the 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 OC} in two different ways:

  • Using the triangle AOC gives
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=b\sin(A)}  ;
  • and using the triangle BOC gives
.
  • Eliminate 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 these two equations:
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\sin(B)=b\sin(A)} .
  • Rearrange to obtain
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 \frac{a}{\sin(A)}=\frac{b}{\sin(B)}}

By using the other two perpendiculars the full law of sines can be proved. QED.

Application

This formula can be used to find the other two sides of a triangle when one side and the three angles are known. (If two angles are known, the third is easily found since the sum of the angles is 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 180^\circ} .) See Solving Triangles Given ASA. It can also be used to find an angle when two sides and the angle opposite one side are known.

Area of a triangle

The area of a triangle may be found in various ways. If all three sides are known, use Heron's theorem.

If two sides and the included angle are known, consider the second diagram above. Let the sides 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} , and the angle between them 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 \alpha} be known. The terms /alpha and /gamma are variables represented by Greek alphabet letters, and these are commonly used interchangeably in trigonometry just like English variables x, y, z, a, b, c, etc. From 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 ACO} , the 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=CO} is 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\sin(\alpha)} so the area is 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 \frac{bc\sin(\alpha)}{2}} .

If two angles and the included side are known, again consider the second diagram above. Let the side and the 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 \alpha} and be known. Let 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 AO=x} . Then

Thus

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=\frac{c}{\tan(\alpha)+\tan(\gamma)}\text{ so area }=\frac{c^2}{2\bigl(\tan(\alpha)+\tan(\gamma)\bigr)}} .


Resources

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