Question

Solve the equation :

`\cos(x) - \sin(x) = √(2 ) \: cos(3x)`
​

Answer 1

General solution is

`x = n \pi + (\pi )/(8)`

Explanation:

Step(i):-

Given cos x - sin x = √2 cos (3 x)

Dividing '√2' on both sides , we get

`(1)/(√(2) ) cos (x) - (1)/(√(2) ) sin (x) = (√(2) cos (3 x))/(√(2) )`

we will use trigonometry formulas

a) Cos ( A + B) = Cos A Cos B - sin A sin B

b)
`cos (\pi )/(4) = (1)/(√(2) )`

Step(ii):-

`(1)/(√(2) ) cos (x) - (1)/(√(2) ) sin (x) = (√(2) cos (3 x))/(√(2) )`

`cos ((\pi )/(4) ) cos x - sin((\pi )/(4) ) sin x = cos 3x`

`cos ((\pi )/(4)+x ) = cos 3 x`

Step(iii):-

General solution of cos x = cos ∝ is x = 2 nπ+∝

we have
`cos ((\pi )/(4)+x ) = cos 3 x`

The general solution of
`cos ((\pi )/(4)+x ) = cos 3 x` is

⇒
`3 x = 2 n \pi + ((\pi )/(4)+x )`

⇒
`3 x- x = 2 n \pi + (\pi )/(4)`

`2x = 2 n \pi + (\pi )/(4)`

final answer:-

General solution is

`x = n \pi + (\pi )/(8)`