The number of values of x lying in the interval (−π, π) which satisfy the equation 8^{1+|cos x|+cos²x+|cos³x|+....∞} = 4³ is:
4
The exponent is the infinite geometric series \(1+|\cos x|+\cos^2 x+|\cos^3 x|+\cdots = \sum_{n=0}^{\infty}|\cos x|^n\), since \(|\cos^n x| = |\cos x|^n\).
This series converges to \(\dfrac{1}{1-|\cos x|}\) (for \(|\cos x|<1\)), so the equation becomes \(8^{1/(1-|\cos x|)} = 4^3 = 2^6\).
Since \(8=2^3\), this gives \(2^{3/(1-|\cos x|)} = 2^6\), so \(\dfrac{3}{1-|\cos x|}=6\), hence \(1-|\cos x| = \dfrac{1}{2}\), giving \(|\cos x| = \dfrac{1}{2}\).
In \((-\pi,\pi)\), \(\cos x = \tfrac{1}{2}\) gives \(x = \pm\pi/3\), and \(\cos x = -\tfrac{1}{2}\) gives \(x = \pm 2\pi/3\).
So there are 4 distinct solutions in \((-\pi,\pi)\).
Similar Questions
Let S be the set of all α ∈ R such that the equation cos(2x) + α sin x = 2α − 7 has a solution. Then S is equal to:
The period of
\(\dfrac{|\sin 4x|+|\cos 4x|}{|\sin 4x-\cos 4x|+|\sin 4x+\cos 4x|}\) is:
Which of the following is greatest?
Important Questions from Trigonometric Functions
If \(\tan A=\dfrac{1}{7}\) , then what is cos 2A equal to?
What is tan 2A equal to?
What is the value of tan 18°?
What is the period of the function f(x) = ln(2 + sin 2x)?
If tan (45° + θ) = 1 + sin 2θ, where \(-\frac{\pi}{4}< θ <\frac{\pi}{4},\) then what is the value of cos 2θ?
