IF `vec(a),vec(b) and vec(c )` represent the three sides of a triangle as shown then using the properties of scalar product and vector product prove that
` . ` B/sinB = C/sinC

Explanation

Since `vec(b)` is the vector sun of `vec(a) and vec(c )` we have
`vec(b) = vec(c ) + vec(a)…(i) `
Taking dot product with `vec(b)` on both sides ( with `vec(c ) + vec(a)` on right side )
`vec(b).vec(b) = (vec (c )+ vec(a)) . (vec(c )+vec(a))`
` = vec(c ) . ( vec(c )+ vec(a)) + vec(a) . (vec(c )+vec(a)) `
` = vec(c ) . vec(c ) + vec(c ) . vec(a) + vec(a) . vec(c ) + vec(a) . vec(a) `
`b^(2) = c^(2) + ca cos theta + ca cos theta + a^(2)`
This is the law of cousines .