# Use the law of cosines to solve the angle V in triangle VGK if KG=25, VG=33,5, <G=46 degrees.

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For a triangle with sides a, b and c opposite to angles A, B and C, the law of cosines gives the relation c^2 = a^ + b^2 -2a*b*cos C.

Here we have KG = 25 and VG = 33.5

So we can find KV^2 = 25^2 + 33.5^2 - 2*25*33.5*cos 46.

=> KV = 24.15

Now, to find the angle V, we have

GK^2 = KV^2 + GV^2 - 2*KV*GV*cos V

=> cos V = -(25^2 - 33.5^2 - 24.15^2)/(2* 33.5*24.15)

=> cos V = .6677

=> V = arc cos .6677

=> V= 48.10 degrees

**The angle V is 48.10 degrees.**

We'll write the law of cosine to determine the angle V:

KG^2 = VG^2 + KV^2 - 2VG*KV*cos V (1)

We notice that the value of KV is missing, but we could find out, applying the law of cosine for the angle G.

KV^2 = GK^2 + GV^2 - 2GK*GV*cos G

KV^2 = 625 + 1122.25 - 1675*cos 46

KV^2 = 625 + 1122.25 - 1163.55

KV^2 = 583.7

KV = 24.15

We'll substitute KV in (1):

625 = 1122.25 + 583.7 - 1618.05*cos V

**cos V = 0.6680**