Can you use the Law of Cosines if you are given a = 41, b = 55 and the measure of angle A equals 56 degrees ? Why or why not?

No you cannot use the law of cosines

Instead you would go with the law of sines. You could write

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

sin(56)/41 = sin(B)/55

and solve for angle B.

The law of cosines only works if you know two sides and the included angle between those sides. So if we know that a = 41 and b = 55, then we can use the law of cosines if we knew the value of angle C.

A slight variation is that we can use the law of cosines if we knew all three sides of the triangle (knowing the angles isn't needed as you can solve for them using the law of cosines).

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