1. ## [SOLVED] Trigonometric equation?

Find the values of x which satisfy the equation 8^(1+│cos x│+ cos^2 x +cos^3 x│+.........infinite terms) = 4^3

^ - raised to
│- Absolute value

Answer: {x│x = 2nπ ± π/3} U {x│x = 2nπ ± 2π/3}

How do we solve this problem?

2. Originally Posted by fardeen_gen
Find the values of x which satisfy the equation 8^(1+│cos x│+ cos^2 x +cos^3 x│+.........infinite terms) = 4^3

^ - raised to
│- Absolute value

Answer: {x│x = 2nπ ± π/3} U {x│x = 2nπ ± 2π/3}

How do we solve this problem?

$\displaystyle 8^{1+|\cos x| + |\cos^2 x| + |\cos^3 x| + ...}$

$\displaystyle 1+|\cos x| + |\cos^2 x| + |\cos^3 x| + ...$ is a geometric series with $\displaystyle r=|\cos x|$

so we have the sum as $\displaystyle \frac{1}{1- |\cos x|}$

and so we have $\displaystyle 8^{\frac{1}{1- |\cos x|}} = 4^3$

$\displaystyle \frac{1}{1- |\cos x|} \ln 2^3 = \ln 4^3 = \ln 2^6$

$\displaystyle 3\frac{1}{1- |\cos x|} \ln 2 = 6\ln 2$

simplifies to $\displaystyle \frac{1}{1- |\cos x|} = 2$

i hope you can finish this already..