Final answer:
Using a graph of f(x) = cos^2(x) - 3/4, we locate the intersection with y = -0.15 to find a number δ: x - 6 < δ where the inequality cos^2(x) - 3/4 < 0.15 holds true. The δ value corresponds to the distance from 6 to this point on the x-axis.
Step-by-step explanation:
To find a number δ such that if x - 6 < δ, then cos2(x) - ¾ < 0.15, we can use a graph of the function f(x) = cos2(x) - ¾. First, we plot the function f(x) and identify the regions where f(x) < 0.15. The value of δ corresponds to the distance along the x-axis between 6 and the point where the function first exceeds the value of -0.15 when moving away from x = 6.
To solve this graphically, we plot the horizontal line y = -0.15 and observe where it intersects the curve f(x). The x-coordinate of this intersection gives us a value that, when subtracted from 6, yields δ. Since no graph is provided here, you will need to sketch or use graphing software to find this intersection. Remember, you may assume data taken from graphs is accurate to three digits.
If we consider the periodic properties of the cosine function, we should look for the first intersection after x = 6. This approach will help us accurately determine the proper δ value that satisfies the given inequality. For our given function, this means identifying the value of x that lies in the appropriate range of the cosine function where the inequality is true.