Explanation:
To find the points of intersection between the two functions, we need to set them equal to each other and solve for x.
1) Setting the two functions equal to each other:
cos^2(x) = sin^2(x) - 1/2
We can use the trigonometric identity cos^2(x) + sin^2(x) = 1 to simplify the equation:
1 - sin^2(x) = sin^2(x) - 1/2
Rearranging the equation:
2sin^2(x) - sin^2(x) = 3/2
sin^2(x) = 3/2
sin(x) = ±√(3/2)
Now we can solve for x by taking the inverse sine of both sides:
x = arcsin(±√(3/2))
2) To state the points of intersection on the interval 0 <= x <= 2π in terms of π, we need to convert the values of x from radians to multiples of π.
Using the unit circle and the symmetry of sine, we know that sin(x) = √(3/2) has two solutions on the interval [0, π]:
x = π/3 and x = 2π/3
Similarly, sin(x) = -√(3/2) has two solutions on the interval [0, π]:
x = 4π/3 and x = 5π/3
Therefore, the points of intersection on the interval 0 <= x <= 2π in terms of π are:
(π/3, y)
(2π/3, y)
(4π/3, y)
(5π/3, y)
Note: The value of y will depend on the specific x-coordinate chosen from the solutions above.