Calculate expression E=a*(b*cosC-c*cosB) if a,b,c are the lengths of the triangle ABC.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Since the type of triangle is not indicated in the given enunciation, we'll consider an acute triangle.

We'll apply cosine theorem in an acute triangle, to express the terms cos C and cos B.

The lengths of the sides of the triangle are: BC = a, AC = b, AB = c.

cos C = (a^2 + b^2 - c^2)/2ab

cos B = (a^2 + c^2 - b^2)/2ac

We'll substitute cos C and cos B into the expression to be calculated.

E = a*[b*(a^2 + b^2 - c^2)/2ab - c*(a^2 + c^2 - b^2)/2ac]

We'll simplify and we'll get:

E = a*[(a^2 + b^2 - c^2)/2a - (a^2 + c^2 - b^2)/2a]

E = a^2/2 + b^2/2 - c^2/2 - a^2/2 - c^2/2 + b^2/2

We'll eliminate like terms and we'll combine the like terms:

E = 2b^2/2 - 2c^2/2

E = b^2 - c^2

The requested value of the expression is represented by the difference of the squares: E = b^2 - c^2.