Prove that in any triangle ABC:

$sin space straight A space equals space 2 over bc square root of straight s left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis end root$

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Solution

sinA equals 2 sin straight A over 2 cos straight A over 2 equals 2 square root of fraction numerator left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator bc end fraction end root square root of fraction numerator straight s left parenthesis straight s minus straight a right parenthesis over denominator bc end fraction end root

equals 2 over bc square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root

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