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.