"Simplify expression 2cos⁡(x)+14cos⁡(x2+π6)" was answered by students like you

Madilynn Avery

Answered question

2022-04-28

Simplify expression

\frac{2 \cos (x) + 1}{4 \cos (\frac{x}{2} + \frac{π}{6})}

Daisy Patrick

Beginner2022-04-29Added 16 answers

Let

\frac{x}{2} + \frac{π}{6} = y

. Then

x = 2 y - \frac{π}{3}

\frac{2 \cos x + 1}{4 \cos (\frac{x}{2} + \frac{π}{6})} = \frac{2 \cos (2 y - \frac{π}{3}) + 1}{4 \cos y}

= \frac{2 \cos 2 y \cos \frac{π}{3} + 2 \sin 2 y \sin \frac{π}{3} + 1}{4 \cos y}

= \frac{2 \cos^{2} y - 1 + 2 \sqrt{3} \sin y \cos y + 1}{4 \cos y}

= \frac{\cos y + \sqrt{3} \sin y}{2}

= \cos y \cos \frac{π}{3} + \sin y \sin \frac{π}{3}

= \cos (y - \frac{π}{3})

= \cos (\frac{x}{2} - \frac{π}{6})

veceritzpzg

Beginner2022-04-30Added 16 answers

Using half-angle formula:

\cos \frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}}

with

A = x + \frac{π}{3}

to give

\cos (\frac{x}{2} + \frac{π}{6}) = \pm \sqrt{\frac{1 + \cos (x + \frac{π}{3})}{2}} = \pm \sqrt{\frac{1 + \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x}{2}}

\cos (\frac{x}{2} + \frac{π}{6}) = \pm \frac{\sqrt{1 + \sqrt{3} \cos x - \sin x}}{2}

Hence

\frac{2 \cos x + 1}{4 \cos (\frac{x}{2} + \frac{π}{6})} = \pm \frac{2 (2 \cos x + 1)}{\sqrt{1 + \sqrt{3} \cos x - \sin x}}

which is positive if

\frac{x}{2} + \frac{π}{6}

is in quadrant I or IV and negative if

\frac{x}{2} + \frac{π}{6}

is in quadrant II or III.

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