Question 1124547: Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B1 is larger than ∠B2.)
a = 33, c = 42, ∠A = 37°
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
For simplicity, I always draw the figure for this problem in the same orientation:
given angle at the left; unknown side length horizontal.
In this example, then, AB (or c) length 42 is slanted up to the right; side BC (or a) is length 33.
The important number here is the height of the triangle, which from the definition of sine is 42*sin(37) = 25.3 to one decimal place. Since side a is longer than 25.3 and less than 42, there will be two triangles.
Then the law of sines tells us
sin(C)/42 = sin(A)/33 --> C = sin^-1(42*sin(37)) = 50 degrees.
In the second triangle angle C is 180-50 = 130 degrees.
Use the fact that the sum of the angles of each triangle is 180 degrees to find the measure of angle B in each triangle.
Use the law of sines with each of those measures of angle B to find the lengths of AC in the two triangles.
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