Math Question?

Question

2 cos 3x - radical 2 = 0

Please explain/show all work.
There should be six answers.

Answer 1

2 cos 3x - rad 2 = 0

2 cos 3x = rad 2

cos 3x = rad 2 / 2

3x = pi/4, 7pi/4, 9pi/4, 15pi/4, 17pi/4, 23pi/4

or

3x = 45 deg, 315 deg, 405 deg, 675 deg, 765 deg, 1035 deg

x = pi/12, 7pi/12, 3pi/4, 5pi/4, 17pi/12, 23pi/12

or

x = 15 deg, 105 deg, 135 deg, 225 deg, 255 deg, 345 deg

(the difference in the last two steps is the top version is in radians and the bottom is in degrees) (i also assumed that the domain was 0

Answer 2

There are actually infinite answers. I will assume that you want answers in the range (0,2pi)

2 cos 3x - sqrt(2) = 0
2 cos(3x) = sqrt(2)
cos(3x) = 1/sqrt(2)
3x = (pi/4, 7pi/4, 9pi/4, 15pi/4, 17pi/4, 23pi/4)

For 0 x = (pi/12, 7pi/12, 3pi/4, 5pi/4, 17pi/12, 23pi/12)
or converting these to degrees
x = (15, 105, 135, 225, 255, 345)

Answer 3

2 cos (3x) - √2 = 0
2 cos (3x) = √2
cos (3x) = √2/2
3x = π/4+2πk ∨ 3x = -π/4+2πk for some k∈Z.
x = π/12+2πk/3 or x=-π/12+2πk/3

This is the complete solution set, but if your teacher expects only six answers, she probably just wants the solutions in some specific range, such as [-π, π]. In this case, we consider only k∈{-1, 0, 1}, since for other integers, the angles obtained will not be in that range. For these values we have:

x∈{-7π/12, π/12, 9π/12, -9π/12, -π/12, 7π/12}

And we are done.

Answer 4

2 cos 3x √2=0
2 cos 3x=√2
cos 3x=√2 /2
3x=arc cos √2 /2
3x=π/4, 7π/4, 9π/4, 15π/4, 17π/4, 23π/4
x=π/12, 7π/12, 3π/4, 5π/4, 17π/12, 23π/12
or
3x=45º, 315º, 405º, 675º, 765º, 1035º
x=15º, 105º, 135º, 225º, 255º, 345º

You didn't specify degrees or radians so I gave you both.

Answer 5

IDK