Question

In triangle ABC, right angle At B if tan A = 1/3 find the values of the following
sin A cos C + cos A sin C
cos A cos C- sin A sin C​

Answer 1

$\bf \LARGE\color{pink}Hello!$

${ \huge{ \star }}\: \: \: \: \Large \underline{ \bf \bold{GiveN :}}$

${ { \rightsquigarrow} }\: \: \: \: \sf \: \triangle \: ABC \: \: is \: \: a \: \: \bf{ right \: \: triangle}$

${ { \rightsquigarrow} }\: \sf \: \: \: \angle \: B = \frac{ \pi}{2}$

${ { \rightsquigarrow} }\: \sf \: \: \tan \: A = \frac{1}{3}$

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${ \huge{ \star}} \LARGE \: \: \: \underline{\bf{To \: \: FinD :}}$

${ \large{\rightsquigarrow}} \: \sf \: \sin A \: \cos C \: + \: \cos A \: \sin C = {?}$

$\sf \rightsquigarrow \: \cos A \cos C- \sin A \sin C = {?}^{}$

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${ \huge{ \star}} \LARGE \: \: \: \underline{\bf{ SolutioN :}}$

As we know, sum of all angles of a triangle is π

So we can say,

$\: \: \: \: \: \: \: \: \sf \angle \: A + \angle B + \angle C = \pi$

$\dashrightarrow \sf \angle \: A + \angle C = \pi - \angle B$

$\dashrightarrow \sf \angle \: A + \angle C = \frac{ \pi}{2} \underline{ \: \: \: \: \: \: \: \: \: \: }(1)$

now,

$\sf \circ \: \: \: \: \: \tan \: A = (\frac{1}{3} )$

$\sf \dashrightarrow A = { \tan}^{ - 1} ( \frac{1}{3} )$

From equation (1)

$\sf \dashrightarrow C= \frac{\pi}{2} - { \tan}^{ - 1} ( \frac{1}{3} )$

we know,

$\boxed{{ \large{\circ \: \: }} \: \sf \: \sin A \: \cos C \: + \: \cos A \: \sin C = { \sin(A +C) } }$

putting value of sin (A+C)

$\sf \dashrightarrow \: \sin (A+C) = \sin \{ \cancel{{ \tan}^{ - 1} ( \frac{1}{3} )} + \frac{\pi}{2} - \cancel{{ \tan}^{ - 1} ( \frac{1}{3} )} \}$

$\: \: \sf \therefore \: \: \: \: { \underline{ \boxed{ \sf{\: \sin (A+C) = 1 }}}}$

we know,

$\boxed{ \sf \circ \: \: \: \: \: \cos \: A \cos C - \sin A \sin C = \cos(A + C)}$

Putting value of cos (A+C)

$\sf \dashrightarrow \: \cos (A+C) = \cos \{ \cancel{{ \tan}^{ - 1} ( \frac{1}{3} )} + \frac{\pi}{2} - \cancel{{ \tan}^{ - 1} ( \frac{1}{3} )} \}$

$\: \: \sf \therefore \: \: \: \: { \underline{ \boxed{ \sf{\: \cos (A+C) = 0 }}}}$

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${ \huge{ \star}} \LARGE \: \: \: \underline{\bf{ RequireD \: \: ConcepT :}}$

${{ {1) \: \: }} \: \sf \: \sin A \: \cos C \: + \: \cos A \: \sin C = { \sin(A +C) } }$

${ \sf 2)\: \: \: \cos \: A \cos C - \sin A \sin C = \cos(A + C)}$

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${ \huge{ \star}} \LARGE \: \: \: \underline{\bf{ConcepT \: \: BoosteR :}}$

${{ {1) \: \: }} \: \sf \: \sin A \: \cos C \: - \: \cos A \: \sin C = { \sin(A - C) } }$

${ \sf 2)\: \: \: \cos \: A \cos C + \sin A \sin C = \cos(A - C)}$

$\sf 3) \: \: \: \tan(A+C) = \frac{ \tan A + \tan C }{1 - \tan A \tan C }$

$\sf 4) \: \: \: \tan(A - C) = \frac{ \tan A - \tan C }{1 + \tan A \tan C }$

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