Question

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$\large{{\underline{\color{green}{\bf{find \: the \: general \: solution \: for \: each \: in \: the \: equation - }}}}}$
➪ Cos 3x + Cosx - Cos 2x = 0

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Answer 1

Step-by-step explanation:

Solution :

(cos 3x + cos) – cos 2x = 0

_____________________

We know that,

$\: \: \: \: \: \: \: \: \: \: \: \cos \: x + \cos \: y \: =2 \cos \: ( \frac{x + y}{2} ) \cos \: \frac{x - y}{2}$

Replacing, x by 3x and y by x

_____________________

$2 \cos \: ( \frac{3x + x}{2} ). \cos \: ( \frac{3x - x}{2} ) - \cos \: (2x) = 0$

$= > 2 \cos \: ( \frac{4x}{2} ). \cos \: ( \frac{2x}{2} )- \cos(2x) = 0$

$= > 2 \cos(2x) . \cos(x) - \cos(2x) = 0$

$= > \cos \: 2x (2 \cos \: x - 1) = 0$

Hence :

cos (2x) = 0

or

(2 cos x – 1) = 0

2 cos x = 1

cos x = ½.

We find general solutions of both separation,

⭐ General Solution for cos (2x) = 0

Given :

• cos (2x) = 0

Thus,

• general solution is

$2x = (2n + 1) \frac{\pi}{2}$

$x = (2n + 1) \frac{\pi}{4}$

• Where n € z.

⭐ General Solution for cos x = ½

Let cos x = cos y. (1)

cos x = ½ (2)

From (3) and (4)

=> cos y = ½

=> cos y = cos π/3

=> cos y = π/3.

⭐ General Solution for cos x = cos y

=> x = 2nπ ± y

• Where n € z

Putting y = π/3

=> x = 2nπ ± π/3

• Where n € z.

Hence :

General solutions is For cos (2x) = 0, x (2n + 1)π/4.

Answer 2

Step-by-step explanation:

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$\large{{\underline{\color{green}{\bf{find \: the \: general \: solution \: for \: each \: in \: the \: equation - }}}}}$

➪ Cos 3x + Cosx - Cos 2x = 0