NIMCET 2017 Question Paper with solutions

Question : 85 of 120

Marks: +1, -0

The value of A that satisfies the equation

a sin A + b cos A = c

is equal to

Solution:

Given:

a \sin A+b \cos A=c

Divide both sides by

\;{1}/{√{ a ^{2}+ b ^{2}}}

, we get

⇒ \;{ a }/{√{ a ^{2}+ b ^{2}}} \sin A +\;{ b }/{√{ a ^{2}+ b ^{2}}} \cos A =\;{ c }/{√{ a ^{2}+ b ^{2}}}

Let

\sin α=\;{ a }/{√{ a ^{2}+ b ^{2}}}

and

\cos α=\;{ b }/{√{ a ^{2}+ b ^{2}}}

⇒ \sin A \sin α+\cos A \cos α=\;{c}/{√{a^{2}+b^{2}}}

⇒ \cos ( A -α)=\;{ c }/{√{ a ^{2}+ b ^{2}}}

⇒ A -α=\cos ^{-1} \;{ c }/{√{ a ^{2}+ b ^{2}}}

⇒ A =scos^{-1} \;{ c }/{√{ a ^{2}+ b ^{2}}}+α

Now,

\tan α=\;{\sin α}/{\cos α}=\;{ a }/{ b }

∴ α=\tan ^{-1} \;{ a }/{ b }

So , A =\cos ^{-1} \;{ c }/{√{ a ^{2}+ b ^{2}}}+\tan ^{-1} \;{ a }/{ b }

=\tan ^{-1}(\;{ a }/{ b })+\cos ^{-1}(\;{ c }/{√{ a ^{2}+ b ^{2}}})

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