Question

14) Prove the cosine rule using the projection rule.​

Answer 1

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Answer 2

To Prove

(a) a = b cos C + c cos B

(b) b = c cos A + a cos C

(c) c = a cos B + b cos A

Explanation:

Using Law of cosines

We have cos A = [ (b² + c² - a²) / 2bc]

               cos B = [(c² + a² - b²) / 2ca]

               cos C = [(a² + b² - c² )/ 2ab]

(a) b Cos C + cs B

   = [(a ²+ b² - c²) / 2ab] + c[(c² + a² - b²)/ 2ca]

   = [(a² + b² - c² + c² + a² + b²)/ 2a]

   = 2a²/ 2a

   = a

(b) c cos A + a cos C

    = c[( b² + c² - a²) / 2bc] + a[( a² +b² - c²) / 2ab]

    = [(b² + c² - a² + b² - c²) / 2b]

    = 2b² / 2b

    = b

(c) c = a cos B + b cos A

    = a [( c² + a² - b²) / 2ca] + b [ (b² + c² - a²) / 2bc]

    =  [(c² + a² - b² + b² + c² - a²) / 2c]

    = 2c² / 2c

    = c

Hence proved

   c = a cos B + b cos A