Question 1120463: Law of sines: sin(A)/a = sin(B)/b = sin(C)/c
In △BCD, d = 3, b = 5, and m∠D = 25°. What are the possible approximate measures of angle B?
A) only 90 degrees
B) only 155 degrees
C) 20 degrees and 110 degrees
D) 45 degrees and 135 degrees
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! in triangle BCD, d = 3, b = 5, angle D = 25 degrees.
d = side opposite angle D
b = side opposite angle B
by the law of sines, d / sin(D) = b / sin(B)
since d = 3 and b = 5 and angle D = 25 degrees, this equation becomes:
3 / sin(25) = 5 / sin(B)
cross multiply to get 3 * sin(B) = 5 * sin(25).
solve for sin(B) to get sin(B) = 5 * sin(25) / 3.
that makes sin(B) = .7043637696.
solve for B to get B = arcsin(.7043637696) = 44.77816685.
that's close to 45 degrees, so it looks like selection D is your correct answer.
but selection D says the angle is 45 degrees and 135 degrees.
that indicates that it's possible for two different value to be for angle B.
this is possible in situations where you have the measure of 2 sides and the measure of an angle that is not between those 2 sides.
when you calculate the angle opposite the side of the angle you have to find, you need to check if that angle can be an obtuse angle as well.
what you do is take 180 and subtract it from that angle.
in this case, you got 180 - 45 = 135.
since the sum of the angles of a triangle is 180, and 135 and 25 is 160, then it is possible that the angle B can be 135 as well.
here's a good reference on what i'm talking about.
it's called the ambiguous case.
https://mathbitsnotebook.com/Geometry/TrigApps/TAUsingLawSines.html
this comes from https://mathbitsnotebook.com/Geometry/Geometry.html
the ambiguous case occurs when the side opposite the angle you know of is shorter than the side opposite the angle you need to find.
selection D is the correct answer because it was possible that angle B could be 45 degrees and it could be 135 degrees.
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