If-sin-2x-a-find-the-simplest-expression-for-cos-x-

Question Number 18830 by Joel577 last updated on 30/Jul/17

If \sin 2 x = a

find the simplest expression for \cos x

Commented by mrW1 last updated on 30/Jul/17

\cos 2 x = \pm \sqrt{1 - a^{2}}

{2 \cos}^{2} x - 1 = \pm \sqrt{1 - a^{2}}

\cos^{2} x = \frac{1 \pm \sqrt{1 - a^{2}}}{2}

\cos x = \pm \frac{\sqrt{1 \pm \sqrt{1 - a^{2}}}}{\sqrt{2}}

Commented by Joel577 last updated on 31/Jul/17

t h a n k y o u v e r y m u c h

Answered by 433 last updated on 30/Jul/17

2 \sin x \cos x = a

4 \sin^{2} x \cos^{2} x = a^{2}

4 (1 - \cos^{2} x) \cos^{2} x = a^{2}

\cos^{2} x = b

4 b - 4 b^{2} = a^{2}

4 b^{2} - 4 b + a^{2} = 0

Δ = 16 - 16 a^{2} = 16 (1 - a^{2})

b = \frac{4 \pm \sqrt{16 \times (1 - a^{2})}}{8} = \frac{1 \pm \sqrt{1 - a^{2}}}{2}

= \frac{1 \pm \sqrt{1 - \sin^{2} (2 x)}}{2} = \frac{1 \pm \cos 2 x}{2}

\cos 2 x = 2 \cos^{2} - 1

b = \frac{1 + \cos 2 x}{2} = \frac{1 + \sqrt{1 - a^{2}}}{2}

Commented by Joel577 last updated on 31/Jul/17

t h a n k y o u v e r y m u c h

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