Non Right Triangle Question!?

Question

Question :
(*=degrees)
Triangle ABC is obtuse, and has the following measurements:

Angle B = 40*
Side 'a' = 75
Side 'b' = 50

How do I solve for angle A??
If all steps could be shown that would be fantastic, i want to learn how to do this properly.

Thanks in advance!

Cheers!

Answer 1

When dealing with triangles and solving for their sides and angles, there are two main laws that come into play: The Law of Sines and Law of Cosines. We will use the Law of Sines here. The Law or Sines read as:

a/(Sin A)= b/(Sin B)=c/(Sin C)

This is used mainly when a side missing and the Angle of that side along with a matching side and angle (for example say side a was missing and angle A and side b and angle B were given).

Our first step to look at the Law of Sines and determine which part of it will be used. Since were missing Angle A, yet giving side a, side b, and Angle B, we will use:

a/(Sin A)=b(Sin B)

Secondly, plug in the values for the givens. Our equation now should be:

75/(Sin A)=50/(Sin 40*)
Key Point: Do Not Solve or simplify anything yet.

The third step is to cross multiply. The equaion should now be:

50(Sin A)=75(Sin 40*)

the fourth step is divide both sides by 50. The Equation now is:

Sin A= 75(Sin 40*)/50 Tip: Still dont simplify anything yet.

The last step is to use the inverse function of sine (arcsine) to isolate angle A: Our equation now looks like

A= Sin^-1[ 75(Sin 40*)/50]

plug it into a scientific calculator to get your final answer, which should be roughly 74.62 degrees.

Answer 2

A = 74.62 or 105.38 degrees


You can use law of sines.
Sin A / a = Sin B / b
SinA / 75 = Sin(40) / 50
SinA = (75 Sin(40) )/50
A = Sin^-1 ( 0.9641814145)
A = 74.62 or 105.38 degrees

The reason there are two possible values for A is because when finding the angle of a triangle with law of sines, you introduce an ambiguous case, a case where two possible triangles can have the exact same measures for those two sides and that one angle. You can read about it here: http://en.wikipedia.org/wiki/Law_of_sines#The_ambiguous_case

Solving for the rest of the triangle...
Triangle 1
C = 180-40-74.62
C = 65.38 degrees

Sin B / b = Sin C / c
Sin(40) / 50 = Sin(65.38) / c
c = (50 * Sin(65.38))/Sin(40)
c = 70.71

Triangle 2:
C = 180 - 40 - 105.38
C = 34.62 degrees

Sin B / b = Sin C / c
Sin(40) / 50 = Sin(34.62) / c
c = (50 * Sin(34.62))/Sin(40)
c = 44.19

So, Triangle 1:
A = 74.62 degrees
B = 40 degrees
C = 65.38 degrees
a = 75
b = 50
c = 70.71

And Triangle 2:
A = 105.38 degrees
B = 40 degrees
C = 34.62 degrees
a = 75
b = 50
c = 44.19

Answer 3

The law of sines says that sin(X)/x = sin(Y)/y = sin(Z)/z, where X, Y, and Z are angles of a triangle, and x, y, and z are the sides opposite those angles.

sin(40)/50 = sin(A)/75

75 * sin(40)/50 = sin(A)

0.96 = sin(A)
A = invsin(0.96) = 74.6 degrees