Question

How many distinct triangles can be formed for which m∠X = 51°, x = 5, and y = 2? zero one two

Answer 1

From the law of sines, we have:


`\displaystyle{ (\sin \angle X)/(x)= (\sin \angle Y)/(y)`,

where x and y are the sides opposite to angles X and Y, respectively.


Substituting the known values, we have:




`\displaystyle{ (51^(\circ))/(5)= (\sin \angle Y)/(2)`, thus


`\displaystyle{ \sin \angle Y=(\sin 51^(\circ))/(5)\cdot2\approx (0.777)/(5)\cdot2=0.31`.


Using a calculator, we can find that arcsin(0.31)=18 degrees, approximately.

We know that sine of (180-18)=162 degrees is also 0.31. But 162 and 51 degrees add up to more than 180 degrees.

Thus, there is only one triangle that can be formed under these conditions.