find the solutions of the equation

Question

Find the solutions of the equation that are in the interval [0, 2π). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
cos

COS[2X-(pi/3)]=0 

given this would make cos equal zero at pi/2 and 3pi/2 

one would be aprox1.3 but i dont know how to convert this to an exact number

Answer 1

cos(2x - π/3) = 0

 

2x - π/3 = π/2 + kπ, where k = 0, ±1, ±2, ...

 

2x = 5π/6 + kπ

 

x = 5π/12 + kπ/2

 

If k = 0, then x = 5π/12

 

If k = 1, then x = 11π/12

 

If k = 2, then x = 17π/12

 

If k = 3, then x = 23π/12

 

Other values of k give values of x that are not in the interval

[0, 2π).

 

Solution set:  {5π/12, 11π/12, 17π/12, 23π/12}

Answer 2

You are correct; cos(x)=0 when x is (pi)/2 or 3(pi)/2. However...there's this annoying thing: because this is a trigonometric function, those are not the only places where that is true. cos(x) will always equal zero every (pi)/2 + n(pi), in which you can put any integer in for n.

 

So now you find situations in which 2x-(pi)/3 equal (pi)/2+ n(pi). 

2x-(pi)/3 = (pi)/2 + n(pi)

2x = 5(pi)/6 + n(pi)

x = 5(pi)/12 + n(pi)/2

 

Your problem gives the restrictions that x has to be between [0,2(pi)).

By plugging in n=0, 1, 2, 3, I got answers all within your restricted domain. Your solutions are: 5(pi)/12, 11(pi)/12, 17(pi)/12, and one other solution (I don't want to give you everything, I want to make sure you understand the process).

Check, because you should have 4 solutions total, and they will all be the exact numbers you need. (It was  5(pi)/12 that was closest to your approximate solution of 1.3, so you're on the right track)

 

 

 

Answer 3

Now that you know that solution of cos Θ = 0 is Θ = pi/2 + pi*n you will need to solve the equation

2x - (pi/3) =  Θ

that is equivalent to

2x = Θ + pi/3

or

x = (Θ + pi/3)/2 

Let's substitute Θ from the first line above and we will get

x = (pi/2 + pi*n +pi/3)/2 = (5pi/6 +pi*n)/2 = 5pi/12 + pi*n/2

 

When n = 0: x= 5pi/12

When n = 1: x = 5pi/12 + pi/2 = 11pi/12

Wnen n = 2: x = 5pi/12 + 2pi/2 = 17pi/12

When n = 3: x = 5pi/12 + 3pi/2 = 23pi/12

 

All these four solutions are between 0 and 2pi

 

Hope this helps.