Question

find all solutions of the equation 2cos4x=-1 in the interval [0,2pi]

Answer 1

\[2\cos4x=-1\]\[\cos4x=\frac{ -1 }{ 2 }\]We know that cosine is negative in the 2nd and 3rd quadrants. Since cosx = 1/2 at pi/3, we know that it will equal the multiples of this in the 2nd and 3rd quadrants.

Therefore, 2pi/3 and 4pi/3.

But, we must account for the factor of 4 in the angle of cos4x. So we simply divide the angles we found by 4 to account for this:

2pi/3 --> 2pi/12 --> *pi/6*
4pi/3 --> 4pi/12 --> *pi/3*

pi/6 and pi/3 are the answers

Answer 2

thanks

Answer 3

2cos⁡4x=-1
cos⁡〖4x=(-1)/2〗
cos⁡〖4x=〗 cos⁡(π-π/3),cos⁡〖(π+π/3),cos⁡(3π-π/3) 〗,cos⁡〖(3π+π/3),〗 cos⁡(5π-π/3),cos⁡〖(7π-π/3),cos⁡(7π+π/3) 〗
4x=2π/3,4π/3,8π/3,10π/3,14π/3,16π/3,20π/3,22π/3
X=π/6,π/3,2π/3,5π/6,7π/6,4π/3,5π/3,11π/6