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Trigonometrical Ratios of Standard Angles — MATHEMATICS STD 9 — Question

ICSE BoardEnglish MediumSTD 9MATHEMATICSTrigonometrical Ratios of Standard Angles3 Marks
Question
Without using tables, evaluate the following$: \sec 30^\circ \operatorname{cosec} 60^\circ + \cos 60^\circ\sin 30^\circ .$
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Answer

$\sec 30^{\circ} \operatorname{cosec} 60^{\circ}+\cos 60^{\circ} \sin 30^{\circ} .$
$ \cos 30^{\circ}=\frac{\sqrt{3}}{2}$
$ \Rightarrow \sec 30^{\circ}=\frac{2}{\sqrt{3}}$
$ \sin 60^{\circ}=\frac{\sqrt{3}}{2}$
$ \Rightarrow \operatorname{cosec} 60^{\circ}=\frac{2}{\sqrt{3}}$
$ \cos 60^{\circ}=\frac{1}{2}, \sin 30^{\circ}=\frac{1}{2}$
$ \sec 30^{\circ} \operatorname{cosec} 60^{\circ}+\cos 60^{\circ} \sin 30^{\circ}$
$ =\frac{2}{\sqrt{3}} \times \frac{2}{\sqrt{3}}+\frac{1}{2} \times \frac{1}{2}$
$ =\frac{4}{3}+\frac{1}{4}$
$ =\frac{16+3}{12}$
$ =\frac{19}{12}$

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