sin-x-1-cos-x-1-2-x-

Question Number 162776 by john_santu last updated on 01/Jan/22

\frac{\sin x}{1 - \cos x} = \frac{1}{2}

x = ?

Answered by bobhans last updated on 01/Jan/22

2 \sin x + \cos x = 1

\sqrt{5} (\frac{2}{\sqrt{5}} \sin x + \frac{1}{\sqrt{5}} \cos x) = 1

\cos (x - α) = \frac{1}{\sqrt{5}}; α = \arctan (2)

x = \pm \arccos (\frac{1}{\sqrt{5}}) + \arctan (2) + 2 k π, k \in Z

Answered by Rasheed.Sindhi last updated on 01/Jan/22

\frac{\sin x}{1 - \cos x} = \frac{1}{2}; x = ?

2 \sin x = 1 - \cos x

{4 \sin}^{2} x = 1 + \cos^{2} x - 2 \cos x

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

{5 \cos}^{2} x - 2 \cos x - 3 = 0

{5 \cos}^{2} x - 5 \cos x + 3 \cos x - 3 = 0

5 \cos x (\cos x - 1) + 3 (\cos x - 1) = 0

\underset{\underset{(E x t r a n e o u s)}{x = 2 n π}}{\cos x = 1} ∣ \underset{\underset{\underset{x \approx 126.87 + 2 n π}{}}{\overset{}{x = \cos^{- 1} (\frac{- 3}{5})}} + 2 n π}{\cos x = - \frac{3}{5}}

Answered by Berlindo last updated on 02/Jan/22

\frac{2 \sin (x / 2) \cdot \cos (x / 2)}{2 \sin^{2} (x / 2)} = \frac{1}{2}

\cot (x / 2) = 1 / 2 \Leftrightarrow \tan (x / 2) = 2

x / 2 = \tan^{- 1} 2 + π n, n \in Z

x = {2 \tan}^{- 1} 2 + 2 π n, n \in Z

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