Help Me W/ This Answer Please!!!?

Question

Find the solutions of the equation that are in the interval [0, 2*Pi); separate your answers with commas.

cos(3x - Pi/6) = 0

Answer 1

You need to "undo" the cosine. The obvious choice is the Arccos. So:

Arccos(cos(3x - pi/6) = Arccos (0)

3x - pi/6 = pi/2 + n*pi, n in the integers

The "plus n*pi" thing is because there are two places on the unit circle where the cosine is always the same, always 180 degrees (pi radians) apart. So we have here an infinite number of solutions. We'll have to take care of that, but let's solve the rest of the problem first. It's just algebra at this point:

3x = pi/2 + pi/6 + n*pi

3x = 2pi/3 + n*pi

x = 2pi/9 + n*pi/3

Now you start finding values of n that put the result into the range you want. Since your range starts with 0, that would be a good one to begin with. That gives you

x = 2pi/9

Now: if n = -1, you'll have a negative angle, which is disallowed by the range. We know, then, that 0 is the smallest n that will work. Now we move up until we find one that gives a value greater than 2pi and stop. At n = 1, you'll have

x = 2pi/9 + pi/3 = 5pi/9 which works

At n = 2, you'll have

x = 2pi/9 + 2pi/3 = 8pi/9 which works

And so on.

Answer 2

You might try the link below. They have lots of solutions to different math problems. One was close to yours.

Find all solutions??
cost-sin2t=0

cos(t) - 2 sin(t) cos(t) = 0
cos(t) (1 - 2 sin(t)) = 0
either cos t = 0
or
1 - 2 sin(t) = 0
cos(t) = 0
t = pi/2 + n pi
or
1 - 2 sin(t) = 0
sin(t) = 1/2
t = pi/6 + 2n pi
or
t = 5 pi/6 + 2n pi

Answer 3

you know that the cosine of pi/2 and 3pi/2 is zero.

So,
3x-pi/6 = pi/2 is one answer
3x-pi/6 = 3pi/2 is the other answer.
all you have to do is solve for x for each one, so you'll have two answers for x. Seperate them with commas.
x = x1, x2

simplified:
x1 = 2pi/9
x2 = 5pi/9
so,
x = 2pi/9, 5pi/9

you can plug these answers into the original equation to check your answers. Make sure your calculator is set for radians, not degrees.

Answer 4

cosx=0 at x=pi/2 and 3pi/2. So (3x-pi/6)=pi/2 or 3pi/2

x=2pi/9, 5pi/9

to find the rest, add 2pi/3 (b/c the coefficient of x is 3)

x=2pi/9, 5pi/9, 8pi/9, 11pi/9, 14pi/9, 17pi/9