Find All Real Solutions (problem)

  • Thread starter Cod
  • Start date
In summary, the conversation is about solving the equation 2 cos 3x - 1 = 0. The first step is to simplify it to cos 3x = 1/2. Then, the conversation discusses finding the general solution for cos 3x = 1/2 and dividing it by a factor of 3 to find the solutions for cos x = 1/2. The conversation also suggests using a graph or remembering trigonometric values to solve the equation.
  • #1
Cod
325
4
Problem is:

2 cos 3x - 1 = 0


Here's how I've done it so far:

2 cos 3x = 1

cos 3x = 1/2

Now I'm stuck. Where do I go from here? Do I divide the 3 out making it cos x = 1/6? I know how to find all the real solutions once I find what cos x is. I'm just having trouble figure out exactly how to get cos x in that type of problem.



Any help is grealy appreciated.
 
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  • #2
Originally posted by Cod
Do I divide the 3 out making it cos x = 3/2?
I think you meant cos x = 1/6. No, we can't take the 3 out.

I know how to find all the real solutions once I find what cos x is. I'm just having trouble figure out exactly how to get cos x in that type of problem.
let y = 3x
cos y = 1/2
y = [the general solution]
3x = [the general solution]
x = 1/3 * [the general solution]
 
  • #3
just let me make the [general solution] a little bit clearer.
Now, we first define y=3x
therefore :
cos y = 1/2
or :
y = cos-1(1/2) + 2n[pi]
Now, you might be thinking "Where did 2n[pi] come from ?"
Well, you must remember that Cos() repeats itself after 2[pi], 4[pi], 6[pi], ... , or in other words after 2n[pi] (where n is an integer), so the value of cos(z) and cos(z+2n[pi]) is the same.
Now, from the last equation, you must either use a calculator to figure out cos-1(1/2), or try to remember if the number looks familiar.
The number looks familiar to me, i know that Cos([pi]/3) = 1/2 , or
Cos-1(1/2) = [pi]/3
So, now, you can rewrite your equation, and solve it.
 
  • #4
I think it would be quite useful if you can draw a y = cos x graph.


Anyway, you just need to find all the solutions where cos x = 1/2. Then divide x by a factor of 3.
 

1. What is the definition of "real solutions" in this context?

Real solutions refer to the values of the variable(s) in a mathematical equation or system of equations that make the equations true. In other words, these are the values that satisfy the equation(s) when plugged in.

2. How do I know if an equation has real solutions?

An equation has real solutions if there exists at least one value for the variable(s) that makes the equation true. This can be determined by solving the equation or system of equations and checking if any real numbers are obtained as solutions.

3. What is the process for finding all real solutions to an equation?

The process for finding all real solutions to an equation involves manipulating and simplifying the equation(s) to isolate the variable(s) on one side and constants on the other. Then, solve for the variable(s) using appropriate mathematical operations. It is important to check the solutions obtained to ensure they are valid and satisfy the original equation(s).

4. Are there any special cases where an equation may not have real solutions?

Yes, there are a few special cases where an equation may not have real solutions. These include equations with imaginary numbers, such as the square root of a negative number, and equations with no solutions, such as a contradiction like 2 = 3. It is important to check for these cases when solving equations.

5. Can an equation have multiple real solutions?

Yes, an equation can have multiple real solutions. This is often the case for higher degree equations, such as quadratic or cubic equations. These equations can have two or three real solutions respectively. It is important to find and consider all possible solutions when solving equations.

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