question_answer

If \[\theta \] is an acute angle and \[4\,\sin \theta =3\], then the value of \[4\,{{\sin }^{2}}\theta -3\,{{\cos }^{2}}\theta +2\]is

A) \[\frac{45}{16}\]

B) \[\frac{35}{18}\]

C) \[\frac{32}{16}\]

D) \[\frac{47}{16}\]

Correct Answer: D

Solution :

Given, \[4\,\sin \theta =3\]

\[\Rightarrow \,\,\,\sin \theta =\frac{3}{4}=\frac{Opposite\,side}{Hypotenuse}\]

Let \[AB=3k\]and \[AC=4k\]

Apply Pythagoras theorem,

\[A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,{{\left( 3k \right)}^{2}}+B{{C}^{2}}={{\left( 4k \right)}^{2}}\]

\[\Rightarrow \,\,\,\,\,BC=\left( \sqrt{16-9} \right)k\]

\[=\sqrt{7}\,\,\,k\]

\[\Rightarrow \,\,\,\cos \theta =\frac{\sqrt{7}k}{4k}=\frac{\sqrt{7}}{4}\]

\[\Rightarrow \,\,\,\,\,\,4{{\sin }^{2}}\theta -3{{\cos }^{2}}\theta +2=4\times \frac{9}{16}-3\times \frac{7}{16}+2\]

\[=\frac{36-21+32}{16}=\frac{47}{16}\]