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# Law of Sines/Ambiguous Case

by

## Tara Pal

on 18 June 2014

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#### Transcript of Law of Sines/Ambiguous Case

The Law of Sines states that the ratio of a side length to its corresponding angle is constant in a triangle.
What is the Law of Sines?
Law of Sines can be useful to you in the same way that the Law of Cosines can be used: to solve for missing dimensions and angles of a triangle. However, the Law of Sines is not always the best choice; sometimes it is better to use the Law of Cosines.
How Can I Use the Law of Sines?
Bibliography
The Law of Sines works best when used to find a missing side or sides of triangles with several given dimensions. For example, AAS triangles (two angles and one opposite side given) make it simple to solve for missing sides with the Law of Sines because angles are already given.
When using the Law of Sines to solve for an angle measure, arcsin returns two different possible angle measures,creating an Ambiguous Case.
When Should I Use Law of Sines?
Law of Sines/Ambiguous Case
The Ambiguous Case occurs when two sides and one opposite angle are given (ASS). Sometimes, the sides and angle given can belong to up to two different triangles. It is tricky to use the Law of Sines to solve in the Ambiguous Case because of this possible more than one angle measure.

sin(x) = y

For one y value, there are infinite possible values for x (for triangles, there are two because the angles add up to 180˚).
What is the Ambiguous Case?
What do I do with the Ambiguous case?
If the triangle is ASS:
1) solve for the missing side using Law of Cosines
2) multiply bsinA, if a < bsinA there is
no
solution
a = bsinA there is
one
solution
a ≥ b there is
one
solution
and if bsinA < a < b there are
two
solutions
3) check solution: is the largest angle opposite the largest side?
does the solution match what makes sense?
Trig can be a "sine" of rebellion! #dangerous
LAW OF SINES PRACTICE PROBLEM

AMBIGUOUS CASE
PRACTICE PROBLEM
From the triangle given above, use Law of Sines to find x
Substitute the information you already know into the formula for the Law of Sines:
sin(73)/90 = sin(71)/x
cross multiplying: xsin(73) = 90sin(71)
x = (90sin(71))/sin(73)
x = 88.985
QUICK!
solve for x if x = arcsin(1/2)

x = arcsin1/2
x = 60˚
x = - 60˚ or 300˚
There is more than one possible angle measure returned for a single sine value.
determine how many triangles are possible from the given dimensions
bsinA = height of triangle
bsinA
25˚
60 = a, <A = 25˚, b = 100
bsinA = 100sin25˚ = 42.262
60 > 42.626
a > b, and when a ≥ b there is
one solution
ONE MORE AMBIGUOUS CASE PRACTICE PROBLEM
In triangle ABC, a = 10, b = 15.5, and angle A = 40˚.
Find all possible values of side c.
How do I make sense of and remember these rules?
Remember to think of bsinA as the height of the triangle. This will help you to understand the relativity between bsinA and a. For example, if side a measured less than the height of the triangle, the shortest distance from the bottom to the top of the triangle, it would not connect with side c, as shown in the diagram below.
11.02
12.73
Gang Sines. Digital image. Web. 4 June 2014. <http://media-cache-ec0.pinimg.com/originals/53/9e/38/539e38d08d637aa65143cd2a8b70fc5c.jpg>.
Math.jpg (700 X 583). Digital image. Web. 2 June 2014. <http://84d1f3.medialib.glogster.com/media/a1/a1e164ca567afcef201e07146407a1737e1b0dc3f598ced07e4effa7da547a12/math.jpg>.
Protesting against ambiguous cartoons. Digital image. Bizarro Comics. Web. 4 June 2014. <http://bizarrocomics.com/files/uploads/2012/06/Bz-04-09-99-ambiguousWEB.jpg>.
"Δ Side, Angle, & Area Calculator." Triangle Calculator. Web. 10 June 2014. <http://www.had2know.com/academics/sas-asa-sss-triangle-calculator.html>.
Sinx-with-arcsin-0.3-both.jpg (389x242). Digital image. Brain Jammer. Web. 4 June 2014. <http://www.brainjammer.com/math/getting-all-inverse-trig-function-values/sinx-with-arcsin-0.3-both.jpg>.
Solving for SAS. Digital image. Math Is Fun. Web. 2 June 2014. <http://www.mathsisfun.com/algebra/images/trig-sasex1.gif>.
The_Law_of_Sines_30.gif. Digital image. Web. 8 June 2014. <http://00.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Law_of_Sines_30.gif>.
The_Law_of_Sines_57.gif. Digital image. Web. 8 June 2014. <http://00.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Law_of_Sines_57.gif>.
TriangleABC.gif (334x226). Digital image. 2000 Clicks. Web. 4 June 2014. <http://2000clicks.com/mathhelp/TriangleABC.gif>.
Unit Circle. Digital image. Web. 8 June 2014. <http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215_lg.gif>.
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