Solve each equation in the interval (0,2pie)

Question

3sec^2x-4=0

Answer 1

Zach,

 

Solve the equation for sec 2x:

 

3 sec² 2x = 4

 

sec² 2x = 4/3

 

sec 2x = ± 2/√3 = ± (2√3)/3 .

 

Depending on how you were taught, I could explain this at least two ways, but here is one.

 

You are looking for all solutions in the interval (0, 2π). Since this is an interval for x, and your equation involves the angle 2x, you can solve for 2x in the interval (0, 4π), then divide all of the solutions by 2 to solve for x.

 

Here is how it should go:

 

sec 2x = (2√3)/3 means cos 2x = √3/2. This can occur only in quadrants I and IV, each with a reference angle of π/6. This means there is a solution of π/6 and also of 2π - π/6 = 11π/6. You can get two more solutions by adding 2π to each of this to fill out the interval from 0 to 4π. These solutions will be 13π/6 and 23π/6.

 

You should try to find the solutions to sec 2x = -(2√3)/3 in the same way.

 

Try it and reply back if you have any problems with this. There should be a total of eight solutions for 2x. The last step will be to divide EACH of them by 2 to get the solutions for x.

 

Based upon what we have already found, you will have the four solutions:

 

x = π/12, 11π/12, 13π/12, and 23π/12.

 

There will be four more to round out the eight total solutions.