Question

evaluate sin5pi/12 . cospi/12 using(sin(a+b)+sin(a-b))

Answer 1

TO FIND :–

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• Value of $\: \: \bf \sin \left( \dfrac{5 \pi}{12} \right) + \cos \left( \dfrac{ \pi}{12} \right) = ? \: \:$

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SOLUTION :–

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$\: \: \implies \bf \sin \left( \dfrac{5 \pi}{12} \right) + \cos \left( \dfrac{ \pi}{12} \right) \: \:$

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• Using identity –

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$\: \: \bf \to \: \: \sin(A+B)+ \sin(A-B) = 2 \sin(A) \cos(B) \: \:$

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• Here –

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$\: \: \implies \bf \: A+B = \dfrac{5\pi}{12} \: \: \: \: \: - - - - eq.(1)$

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• And –

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$\: \: \implies \bf \: A - B = \dfrac{\pi}{12} \: \: \: \: \: - - - - eq.(2)$

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• Add both equation –

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$\: \: \implies \bf \:A+B+A-B = \dfrac{5\pi}{12} + \dfrac{\pi}{12}$

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$\: \: \implies \bf \:2A = \dfrac{6\pi}{12}$

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$\: \: \implies \bf \:2A = \dfrac{\pi}{2}$

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$\:\: \implies \large { \boxed{ \bf \:A = \dfrac{\pi}{4}}}$

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• Now Using eq.(1) –

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$\: \: \implies \bf \: \dfrac{\pi}{4} +B = \dfrac{5\pi}{12}$

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$\: \: \implies \bf \: B = \dfrac{5\pi}{12} - \dfrac{\pi}{4}$

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$\: \: \implies \large { \boxed{\bf \: B = \dfrac{\pi}{6}}}$

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• Now Using formula –

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$\: \: \implies \bf \sin \left( \dfrac{5 \pi}{12} \right) + \cos \left( \dfrac{ \pi}{12} \right) = 2 \sin \left( \dfrac{\pi}{4} \right)\cos \left( \dfrac{\pi}{6} \right)\: \:$

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$\: \: \implies \bf \sin \left( \dfrac{5 \pi}{12} \right) + \cos \left( \dfrac{ \pi}{12} \right) = 2 \left( \dfrac{1}{ \sqrt{2} } \right) \left( \dfrac{ \sqrt{3} }{2} \right)\: \:$

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$\: \: \implies \large{ \boxed{ \bf \sin \left( \dfrac{5 \pi}{12} \right) + \cos \left( \dfrac{ \pi}{12} \right) = \dfrac{ \sqrt{3} }{ \sqrt{2} }}} \: \:$

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▪︎ Hence , The value of $\: \: \bf \sin \left( \dfrac{5 \pi}{12} \right) + \cos \left( \dfrac{ \pi}{12} \right)\:\:$ is $\:\: \sqrt{ \dfrac{3}{2} } \: \:$

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