Correct methodology for solving this trigonometric equation I want the correct method for solving this trigonometric equation sqrt(cos(x) sqrt(cos(x) sqrt(cos(x)...)))=1.

miklintisyt

miklintisyt

Answered question

2022-10-27

Correct methodology for solving this trigonometric equation
I want the correct method for solving this trigonometric equation
cos ( x ) cos ( x ) cos ( x ) = 1.
I have used my own technique:
cos ( x ) cos ( x ) cos ( x ) = 1 cos 1 2 ( x ) cos 1 4 ( x ) cos 1 8 ( x ) = 1 log ( cos 1 2 ( x ) cos 1 4 ( x ) cos 1 8 ( x ) ) = log 1 log ( cos 1 2 ( x ) ) + log ( cos 1 4 ( x ) ) + log ( cos 1 8 ( x ) ) + = log 1 1 2 log ( cos ( x ) ) + 1 4 log ( cos ( x ) ) + 1 8 log ( cos ( x ) ) + = log 1 log ( cos ( x ) ) ( 1 2 + 1 4 + 1 8 + ) = log 1
The series in the brackets is an infinite geometric series by summing it I get
cos ( x ) 1 2 1 1 2 = 1 cos ( x ) 1 2 1 2 = 1 cos ( x ) = 1 x = arccos ( 1 ) x = 0
So the solution set is { 2 n π }.
This is how I have solved the given problem but I want is there any other method for solving this equation.

Answer & Explanation

Steinherrjm

Steinherrjm

Beginner2022-10-28Added 12 answers

For any A > 0,
A = cos x cos x cos x = A cos x cos x = A .
In your case, A = 1, so cos x = 1, so x { 0 , ± 2 π , ± 4 π , }

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