I'm confused with this concept:

Okay, so I have this problem:

cos2x( 2 cos+1)= 0

I understand I have to work the problems separately and I get cos x= 2 and cos x= 1/2

Now, what I don't understand is how this connects to pi, radians or the unit circle. If someone could explain or point me to a place with a thorough explanation, please tell me. B/c trust me, I've looked EVERYWHERE and my book/teacher doesn't really explain the theory very well.

I get cos x= 2 ?????

I do not
I get
cos 2 x = 0
that means that 2x = pi/2 or 90 degrees or 2x = 3 pi/2 or 270 degrees
or x = pi/4 or 3pi/4
then
cos x = -1/2
well cos of 60 degrees or pi/3 is 1/2
so cos (180-60) = -1/2 so x =2pi/3
or 180+60 which is 5pi/3

sry it's cos^2x

I understand your confusion. Let's take a step-by-step approach to understand how the solutions to the equation cos2x(2cosx+1) = 0 are connected to radians and the unit circle.

1. Start with the equation: cos2x(2cosx+1) = 0.

2. To solve this equation, we need to find the values of x that satisfy the equation.

3. First, consider the equation cos2x = 0. This equation means that the cosine of 2x is equal to 0.

4. Recall that the cosine function represents the x-coordinate of a point on the unit circle.

5. On the unit circle, the cosine is equal to 0 at certain angles. In terms of radians, these angles are π/2, 3π/2, 5π/2, etc.

6. To find the possible values of x, we need to determine what value of 2x corresponds to these angles. So, we can set up the equation:

2x = π/2, 3π/2, 5π/2, etc.

Solving for x, we get:

x = π/4, 3π/4, 5π/4, etc.

These are the values of x that satisfy cos2x = 0.

7. Next, consider the equation 2cosx+1 = 0. This equation means that 2 times the cosine of x, plus 1, is equal to 0.

8. Solve this equation:

2cosx+1 = 0

2cosx = -1

cosx = -1/2

Now, consider the cosine function. The cosine is equal to -1/2 at certain angles on the unit circle. In terms of radians, these angles are 2π/3, 4π/3, etc.

9. To find the possible values of x, we need to determine what angles correspond to cosx = -1/2. So, we can set up the equation:

x = 2π/3, 4π/3, etc.

These are the values of x that satisfy cosx = -1/2.

So, the solutions to the original equation cos2x(2cosx+1) = 0 are x = π/4, 3π/4, 5π/4, 2π/3, 4π/3, etc. These values are connected to radians and the unit circle because we are using the properties of the cosine function, which corresponds to the x-coordinate on the unit circle, to find the values of x that satisfy the equation.

For further explanation and practice, you can refer to textbooks or online resources on trigonometry, specifically on solving trigonometric equations. Online tutorials and videos can also be helpful in visualizing the concept.