question_answer

If \[\sec \theta =\frac{25}{7},\] then \[\sin \theta =\]

A) \[\frac{7}{24}\]

B) \[\frac{24}{7}\]

C) \[\frac{24}{25}\]

D) \[\frac{25}{24}\]

Correct Answer: C

Solution :

[c] Let ABC be the right angle triangle such that

\[\angle B=90{}^\circ \] and \[\angle A=\theta \]

\[\therefore \,\,\sec \theta =\frac{AC}{AB}=\frac{25}{7}\]

Let \[AC=25k\] and \[AB=7k,\]

where k is positive constant.

\[\therefore \,\,\,\,\,B{{C}^{2}}=(A{{C}^{2}}-A{{B}^{2}})={{(25k)}^{2}}-{{(7k)}^{2}}\]

\[=(625{{k}^{2}}-49{{k}^{2}})=576{{k}^{2}}\]

\[\Rightarrow \,\,BC=24k\]

\[\therefore \,\,\,\,\sin \theta =\frac{BC}{AC}=\frac{24k}{25k}=\frac{24}{25}\]