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Question Number 10696 by okhema last updated on 23/Feb/17

g i v e n t h a t t a n 2 x = \frac{1}{4} a n d t h a t a n g l e x i s a c u t e, c a l c u l a t e, w i t h o u t u s i n g a c a l c u l a t o r t h e v a l u e o f t h e s e .

(a) c o s 2 x

(b) s i n x

Answered by mrW1 last updated on 23/Feb/17

∴ \tan 2 x = \frac{1}{4} < 1

∵ 2 x < 45 ° a n d x < 22.5 °

(a)

\cos^{2} 2 x = \frac{1}{1 + \tan^{2} 2 x} = \frac{1}{1 + {(\frac{1}{4})}^{2}} = \frac{16}{17}

\cos 2 x = \sqrt{\frac{16}{17}} = \frac{4}{\sqrt{17}}

(b)

\sin^{2} x = \frac{1 - \cos 2 x}{2} = \frac{1 - \frac{4}{\sqrt{17}}}{2} = \frac{\sqrt{17} - 4}{2 \sqrt{17}}

\sin x = \sqrt{\frac{\sqrt{17} - 4}{2 \sqrt{17}}} = \frac{\sqrt{34 - 8 \sqrt{17}}}{2 \sqrt{17}}

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