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Topic: Law of sines


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  Law of Sines   (Site not responding. Last check: 2007-09-10)
The law of sines for plane triangles was known to Ptolemy and by the tenth century Abu'l Wefa had clearly expounded the spherical law of sines.
It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase.
With this convention, the spherical law of sines states that in a spherical triangle with sides a, b, and c and angles A, B, and C, it is true that
www.pballew.net /lawofsin.html   (421 words)

  
 Illuminations: Law of Sines and Law of Cosines
Illuminations: Law of Sines and Law of Cosines
Therefore, the law of sines cannot be used to determine the measures of the missing angles in the triangle with only three sides given.
Therefore, the law of sines cannot be used to determine the measures of the missing angles and side in the triangle with the given sides and included angle, because the side opposite the given angle is unknown.]
illuminations.nctm.org /LessonDetail.aspx?ID=L704   (1592 words)

  
 Law of sines - Wikipedia, the free encyclopedia
In trigonometry, the law of sines (or sine law) is a statement about arbitrary triangles in the plane.
When using the law of sines to solve triangles, under special conditions there exists an ambiguous case where two separate triangles can be constructed (i.e., there are two different possible solutions to the triangle).
Simplifying, the sine of angle A is equal to 5/7, or approximately 0.714.
en.wikipedia.org /wiki/Law_of_sines   (600 words)

  
 Law of Sines   (Site not responding. Last check: 2007-09-10)
The law of sines is used to find angles of a general triangle.
If two sides and the enclosed angle are known, it can be used in conjunction with the law of cosines to find the third side and the other two angles.
It has application along with the law of cosines to the problem of the heading angle for an aircraft in the wind.
hyperphysics.phy-astr.gsu.edu /hbase/lsin.html   (140 words)

  
 Oblique Triangles
These are called the "law of cosines" and the "law of sines." There are other "laws" that used to be used, but since the common use of calculators, these two laws are enough.
The law of cosines relates the three sides of the triangle to one of the angles.
Determine PA using the law of sines for triangle PAB, and determine QA using the law of sines for triangle QAB.
www.clarku.edu /~djoyce/trig/oblique.html   (1604 words)

  
 Law of Cosines
The law of cosines for calculating one side of a triangle when the angle opposite and the other two sides are known.
It has application along with the law of sines to the problem of the heading angle for an aircraft in the wind.
With that angle and the law of sines, the offset angle
hyperphysics.phy-astr.gsu.edu /hbase/lcos.html   (250 words)

  
 The law of sines, including the ambiguous case.
WE USE THE LAW OF SINES AND THE LAW OF COSINES to solve triangles that are not right-angled.
Now, according to the Law of Sines, in every triangle with those angles, the sides are in the ratio 643 : 966 : 906.
But the sine of an angle is equal to the sine of its supplement.
www.themathpage.com /aTrig/law-of-sines.htm   (944 words)

  
 The Laws of Sines and Cosines
The Law of Sines establishes a relationship between the angles and the side lengths of
The essence of the Law of Cosines has been known to Euclid, who proved the obtuse case as II.12 and the acute case as II.13.
Of many other available proofs of the Law of Cosines, a proof without words is a direct generalization of Thâbit ibn Qurra's proof of the Pythagorean proposition.
www.cut-the-knot.org /pythagoras/cosine2.shtml   (734 words)

  
 The Law of Sines
According to a theorem from geometry, the ratio of a side of a triangle to the sine of the opposite angle equals the diameter of the circumscribed circle.
The Law of Sines, together with the Law of Cosines which we will study shortly, are useful for solving triangles.
In order to apply the Law of Sines, it is necessary to know the size of at least one side and the size of the angle opposite that side.
jwbales.home.mindspring.com /precal/part6/part6.1.html?o=0   (882 words)

  
 Law of Cosines
The law of cosines can be used when two sides and the included angle are known or when all three sides are known.
You can use the law of cosines to solve the ambiguous case that we looked at with the law of sines.
The trick is to write the law of cosines as a quadratic in terms of the unknown side.
www.lhs.logan.k12.ut.us /~rweeks/trig/law_of_cosines.htm   (316 words)

  
 Law of Cosines and Law of Sines for Triangles   (Site not responding. Last check: 2007-09-10)
Law of Cosines and Law of Sines for Triangles
Laws of Cosines and Sines for Angles/Sides of Triangles
Enter two angles and any side, or 3 sides, or two sides and angle between them, into the boxes above to get the other sides and angles using the laws of cosines or sines and sum of angles of triangle.
www.wcrl.ars.usda.gov /cec/java/triangle.htm   (96 words)

  
 Oblique Triangles
The Law of Sines is actually a proportion of the side to the sine of its angle.
From that point proceed to solve the triangle with the Law of Sines.
There exists what is called the ambiguous case for the Law of Sines.
www.cgtcollege.org /mat104/lawofsines.html   (391 words)

  
 The (Complete) Law of Sines
One is the connection between sine and the distance to the circumscribed center from the midpoint of each side.
As a side of the triangle approaches the center, its length approaches 2r, of course, so of course the value of sine approaches 1 as the side gets closer to the center and becomes a diameter.
In a circle of this size, the actual lengths of the sides of an inscribed triangle are equal to the sine of the angle opposite that side.
www.valleyview.k12.oh.us /vvhs/dept/math/lawosines.html   (271 words)

  
 LESSON #7   (Site not responding. Last check: 2007-09-10)
For some basic examples using the law of sines and the law of cosines, try aleph0.clarku.edu/~djoyce/java/trig/oblique.html.
If S stands for side length and A stands for angle measure, then the law of sines is used to solve a triangle in the cases SSA, ASA, and AAS and the law of cosines is used in the cases SSS and SAS.
If the value of sine arrived at using the law of sines is greater than one, then there is no solution.
www.hcc.hawaii.edu /distance/math140/lesson7.htm   (514 words)

  
 PlanetMath: sines law proof
implies their sines are the same and so
Q.E.D. "sines law proof" is owned by drini.
This is version 5 of sines law proof, born on 2001-11-11, modified 2002-07-28.
planetmath.org /encyclopedia/SinesLawProof.html   (106 words)

  
 sine law, ambiguous case, concrete representation
The picture at the top left, because it is accurately drawn and labeled using standard pictorial notation (such as the square to indicate right angle), is an example of what might be copied in place of concrete representation.
But, the goal here is to provide a model for concrete representation of an idea -- the ambiguous case of the sine law, finding the angle when a side-angle pair and the unpaired side is given.
This model, made with a transparency film (produced from the master) and 3 number 8 clothing snaps, permits the user to rotate the arm containing side a as desired.
www.mathnstuff.com /math/spoken/here/2class/330/sinelaw.htm   (337 words)

  
 Laws of Cosines & Sines
We saw in the section on oblique triangles that the law of cosines and the law of sines were useful in solving for parts of a triangle if certain other parts are known.
An explanation of the law of sines is fairly easy to follow, but in some cases we'll have to consider sines of obtuse angles.
But the sine of an obtuse angle is the same as the sine of its supplement.
www.clarku.edu /~djoyce/trig/laws.html   (912 words)

  
 The educational encyclopedia, mathematics, trigonometry
Law of cosines law of cosines, to solve triangles that are not right-angled.
Law of sines the law of sines and applications to solving triangles
Law of sines law of sines, to solve triangles that are not right-angled
www.educypedia.be /education/mathematicstrigonometry.htm   (303 words)

  
 Lesson   (Site not responding. Last check: 2007-09-10)
All: Given two angles and a side, every student should be able to find the remaining angle, using their knowledge of a triangle’s angle sum (180 degrees or pi), and the two remaining sides, using the law of sines.
Every student should be able to state the law of sines from memory.
Some students should understand how the law of sines is obtained from the recently learned area formulas.
www.tcnj.edu /~bliszcz2/lesson8.htm   (862 words)

  
 Law of Sines
The Law of Sines is a/(sin A) = b/(sin B) = c/(sin C) = the diameter of the circumscribed circle.
Since the measure of C either equals the measure of D or is supplementary to D, their sines are equal, so c/(sin D) = c/(sin C) = the diameter of the circle.
Laws of Sines and Cosines - and using these laws to find the missing sides and angles in a triangle
mcraefamily.com /MathHelp/GeometryLawOfSinesProof.htm   (290 words)

  
 Sine, Cosine, and Ptolemy's Theorem
ABC we get the Law of Sines which in the standard notations appear as
From the definition of sine and cosine we determine the sides of the quadrilateral.
The Law of Sines supplies the length of the remaining diagonal.
www.cut-the-knot.org /proofs/sine_cosine.shtml   (347 words)

  
 Algebra II: Equations and Triangles - Math for Morons Like Us
When you know two sides and an angle opposite one of the sides, the law of sines can be used.
There are three rules that make up the law of cosines, but you only need to memorize one because the other two can be obtained by changing the letters (put b in place of a, for example).
Secondly, you can use the law of cosines when two sides and the included angle are known.
library.thinkquest.org /20991/alg2/eqtri.html   (871 words)

  
 Oblique Triangles
With these two triangles you realize it would be impossible to use the Law of Sines because we have no pair and no way to get a pair.
We now have a pair so we can use the Law of Sines (beware of the ambiguous case) to finish solving the triangle or we can stick with the Law of Cosines.
We now have a pair so we can go back to the Law of Sines (again beware of ambiguous case) or we can continue with the Law of Cosines.
www.cgtcollege.org /mat104/lawofcosines.html   (322 words)

  
 Math Forum - Problems Library - Trig/Calculus, Law of Sines
The Law of Sines can be thought of as an extension of basic right triangle trig ratios.
More information about the Law of Sines can be found in the Ask Dr. Math archives.
A derivation of the Law of Sines and an explanation of solving triangles using the Law of Sines are examples of what you can find when exploring the High School Trigonometry area of the Ask Dr. Math archives.
mathforum.org /library/problems/sets/trigcalc_lawofsines.html   (321 words)

  
 Law of Sines and Cosines   (Site not responding. Last check: 2007-09-10)
Thus the law of cosines, a generalization of the Pythagorean theorem, is valid when angles are defined as above.
Given the law of cosines, prove the law of sines by expanding sin(θ)
The exact value depends on the shape of the triangle, but for any triangle, the sine of an angle divided by the length of the opposite side is a fixed ratio.
www.mathreference.com /la,law.html   (296 words)

  
 [No title]
Motivation: The Law of Sines is useful if we know a given angle and its opposite side then given another opposing angle or we can find the side opposite or vice versa, given another side we can find the angle opposite.
That is, do something like the proof of the law of sines but for specific sides and angles.
Formal Concept or Definition: The Law of Sines Sketch a general triangle as on page 60 of the text:  B c a h  A C b P (Here we are letting A, B, and C represent both points and their corresponding angles.) Prove the Law of Sines.
math.la.asu.edu /~surgent/mat170/lawofsines.doc   (879 words)

  
 Law of Sines & Law of Cosines
You and your team are assigned the task of designing a web page to help yourselves and other students learn about the 'Law of Sines'.
This means 10 for the Law of Sines.
State the what topics you learned last year in Sequential II and the year before in Sequential I, that are necessary knowledge in order to you to learn and understand the 'Law of Sines' topic.
www.bellmore-merrick.k12.ny.us /webquest/math/sines.html   (771 words)

  
 Common Trigonometry Mistakes Example: Law of Sines
Apply the Law of Sines a second time to find c:
The key in problems using the Law of Sines is to remember that there are two angles between 0° and 180° whose sines have the same value.
If θ is an angle in the first quadrant, then the angle in the second quadrant with the same value for the sine function is 180° - θ.
mathmistakes.info /mistakes/trig/Examples/16/ctm.html   (309 words)

  
 Law of Sines - Key Curriculum Press
The law of sines relates the lengths of the sides of a triangle to the sines of the angles of the triangle.
Use the sine function to express h_b in terms of b and A.
A triangle is shown along with the measurements of its angles in degrees, its side lengths, and the common quotients expressed in the law of sines.
www.keypress.com /x7192.xml   (298 words)

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