The Law of Sines - Revision history

Khanh: /* Resources */

2021-10-28T22:57:31Z

Resources

Khanh at 22:56, 28 October 2021

2021-10-28T22:56:30Z

Lila at 23:06, 7 October 2021

2021-10-07T23:06:50Z

Lila at 23:05, 7 October 2021

2021-10-07T23:05:36Z

Lila at 23:04, 7 October 2021

2021-10-07T23:04:31Z

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T22:19:34Z

fixing link to go straight to pdf

Johnraymond.yanez: Added example

2020-08-12T13:19:00Z

Added example

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

2020-08-11T21:58:22Z

added The Law of Sines notes by professor Esparza

New page

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

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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://www.youtube.com/watch?v=1WXhKIK8oEM Application of the Law of Sines]. Produced by Professor Zachary Sharon, UTSA.

← Older revision		Revision as of 22:56, 28 October 2021
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	* [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.
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Trigonometry/Law_of_Sines Law of Sines, Wikibooks] under a CC BY-SA license

@@ Line 38: / Line 38: @@
 ==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://www.youtube.com/watch?v=1WXhKIK8oEM Application of the Law of Sines]. Produced by Professor Zachary Sharon, UTSA.

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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
 :<math>\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}</math>
-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:
+Each of these expressions is also equal to the diameter of the triangle's circumcircle (the circle that passes through the points <math>A,B,C</math>). The law can also be written in terms of the reciprocals:
 :<math>\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}</math>
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 ==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 <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.
+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> ). It can also be used to find an angle when two sides and the angle opposite one side are known.
 ==Area of a triangle==

← Older revision		Revision as of 23:04, 7 October 2021
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		+	[[Image:Law-of-sines1.svg]]
		+
		+	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
		+	:<math>\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}</math>
		+	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:
		+	:<math>\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}</math>
		+
		+	==Proof==
		+	[[Image:Law-of-sines2.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> .
		+	We can calculate the length <math>h</math> of the altitude <math>OC</math> in two different ways:
		+
		+	*Using the triangle AOC gives
		+	:<math>h=b\sin(A)</math> ;
		+	*and using the triangle BOC gives
		+	:<math>h=a\sin(B)</math> .
		+	*Eliminate <math>h</math> from these two equations:
		+	:<math>a\sin(B)=b\sin(A)</math> .
		+	*Rearrange to obtain
		+	:<math>\frac{a}{\sin(A)}=\frac{b}{\sin(B)}</math>
		+
		+	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 <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.
		+
		+	==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 <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> .
		+
		+	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
		+	:<math>\frac{x}{h}=\tan(\alpha)\text{ ; }\frac{c-x}{h}=\tan(\gamma)\text{ ; adding these, }\frac{c}{h}=\tan(\alpha)+\tan(\gamma)</math>
		+	Thus
		+	:<math>h=\frac{c}{\tan(\alpha)+\tan(\gamma)}\text{ so area }=\frac{c^2}{2\bigl(\tan(\alpha)+\tan(\gamma)\bigr)}</math> .
		+
		+
		+	==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.

← Older revision		Revision as of 22:19, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Sines/ 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.

← Older revision		Revision as of 13:19, 12 August 2020
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Sines/ The Law of Sines]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20Law%20of%20Sines/ 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.