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Fixed spelling, removed hotlinks (#34460)

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Fixed minor spelling/grammar issues, and then removed two of the hotlinked images since they were easy to replace with text instead. Also adjusted the links on the bottom to render on their own lines.
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Alexander Molnar
2019-01-21 12:10:28 -05:00
committed by Tom
parent b862a0fde6
commit 9de2c7da53
1 changed files with 34 additions and 10 deletions
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guide/english/mathematics/law-of-cosines/index.md
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@ -3,23 +3,47 @@ title: Law of Cosines
--- ---
## Law of Cosines ## Law of Cosines
The Law of Cosines can be used to find the ramaining sides or angles of a non-right triangle when given either the length of two sides and its included angle (SAS) or the lengths of the three sides (SSS). The Law of Cosines can be used to find the remaining sides or angles of a non-right triangle when given either the length of two sides and its included angle (SAS) or the lengths of the three sides (SSS).
The Law of Cosines states: The Law of Cosines states:
<img src="http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/lcos.gif"> <img src="http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/lcos.gif">
This looks very similar to the Pythagorean Theorem, and if C was 90 degrees (a right triange), the expression acctually simplifies to the Pythagorean Thoerem. So, the Pythagorean Theorem is a special case of the Law of Cosines. This looks very similar to the Pythagorean Theorem, and if C was 90 degrees (a right triange), then cos(C) = 0 and the expression simplifies to the Pythagorean Thoerem. So, the Pythagorean Theorem is a special case of the Law of Cosines.
You can also switch the angle C with either A or B, but make sure to change the respective side (c) to a or b. You can also switch the angle C with either A or B, but need to change the respective side (c) to a or b:
<img src="http://www.mathwarehouse.com/trigonometry/images/law-of-cosines/law-of-cosines-formula-and-picture2.png"> <p align='center'>
a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2bc &middot; cos(A)<br />
b<sup>2</sup> = a<sup>2</sup> + c<sup>2</sup> - 2ac &middot; cos(B)<br />
c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> - 2ab &middot; cos(C)
</p>
Ex: For example, suppose we have a triangle as above, with b = 21, c = 32 and A = 40&deg;. Then we can find the length of the remaining side a as well as the angles B and C using the cosine law.
<img src="http://www.mathwarehouse.com/sheets/trigonometry/advanced/law-of-sines-and-cosines/images/picture-of-law-of-cosines.png"> To start, let's find a. Since we know b, c and A, the first of the three formulas above will be best to use. We have
<!-- The article goes here, in GitHub-flavored Markdown. Feel free to add YouTube videos, images, and CodePen/JSBin embeds -->
<p align='center'>
a<sup>2</sup> = 21<sup>2</sup> + 32<sup>2</sup> - 2 &middot; 21 &middot 32 &middot; cos(40&deg;)
</p>
so simplifying and taking square roots we have a = sqrt(1465 - 1344 &middot; cos(40&deg;)) or approximately, a &approx; 20.87.
Similarly, if we want to find the angle B, we now have the length of all three sides so the second formula above will be helpful. Rearranging we have
<p align='center'>
cos(B) = [a<sup>2</sup> + c<sup>2</sup> - b<sup>2</sup>]/(2ac)
</p>
and so taking the inverse cosine (or arccosine) of both sides gives the angle B. Plugging in the lengths of our sides (using the exact expression for a found above, and only simplyfing at the end) we find B is approximately 40.31&deg;.
Now to find C we have two approaches, either use the third formula above, or recall that the angles of our triangle should add up to 180 degrees, so if A is 40&deg; and B was found above, we have
<p align='center'>
C = (180 - 40 - B)&deg; &approx; 99.69&deg;
</p>
(Checking with the cosine formula we get the same approximate answer, so we can rest easy knowing we haven't likely made a mistake in any of the calculations.)
#### More Information: #### More Information:
<!-- Please add any articles you think might be helpful to read before writing the article --> <!-- Please add any articles you think might be helpful to read before writing the article -->
https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-cosines - https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-cosines
http://mathworld.wolfram.com/LawofCosines.html - http://mathworld.wolfram.com/LawofCosines.html