Sketch the graphs of y=cos 2x and y= -0.5 over the domain -pi<x<pi. use an algebraic method to determine the values of x where these two graphs intersect. use exact value.

Please help me work this problem out! I sketched the graph, just not sure how to set up the equation. Thankyou

2 x = pi +/- pi/3 radians

you take it from there

To determine the values of x where the graphs of y = cos(2x) and y = -0.5 intersect, we can set up the equation:

cos(2x) = -0.5

Let's solve this equation step-by-step.

Step 1: Convert -0.5 to its equivalent cosine value.
We know that cos(60°) = 0.5, so -0.5 is equivalent to cos(180° + 60°) = cos(240°).

Step 2: Substitute the equivalent cosine value into the equation:

cos(2x) = cos(240°)

Step 3: Apply the identity for cosine of a sum:

2x = ±240° + 2nπ, where n is an integer.

Step 4: Solve for x:

a) For 2x = 240° + 2nπ:
Divide both sides by 2:
x = 120° + nπ, where n is an integer.

b) For 2x = -240° + 2nπ:
Divide both sides by 2:
x = -120° + nπ, where n is an integer.

Now that we have the values of x, we can determine the values of x within the given domain of -π < x < π.

For x = 120°, it falls within the given domain.

For x = -120°, it also falls within the given domain.

So the values of x where the graphs of y = cos(2x) and y = -0.5 intersect within the domain -π < x < π are x = 120° and x = -120°.

To determine the values of x where the graphs y = cos 2x and y = -0.5 intersect, we need to set these two equations equal to each other and solve for x algebraically.

First, let's set up the equation:

cos 2x = -0.5

To solve this equation, we can use the inverse cosine function (also known as arccos or cos⁻¹) to isolate x. The inverse cosine function "undoes" the cosine function, giving us the angle whose cosine is a given value.

Taking the inverse cosine of both sides, we have:

2x = cos⁻¹(-0.5)

Now, we need to find the exact value of cos⁻¹(-0.5). Recall that the cosine function is positive in the first and fourth quadrants, where the x-values are positive, so the angle we are looking for should be in one of these quadrants.

Using the unit circle, we see that cos⁻¹(-0.5) is equal to 120 degrees or 2π/3 radians, which is an angle in the second quadrant.

However, since our initial domain is -π < x < π, we need to find an equivalent angle within that range. To do this, we subtract 2π from 2π/3:

2x = 2π/3 - 2π

Simplifying further:

2x = -4π/3

Dividing both sides by 2:

x = -2π/3

So, the exact value of x where the graphs intersect is -2π/3.