The two points of intersection are: (π/3, 1), (5π/3, 1)
Finding the Intersection Points of Functions
Step 1: Set the functions equal to each other.
We are given two functions:
f(x) = sin(x) * cos(x)
g(x) = 2cos(x)
To find the points where these functions intersect, we set them equal to each other:
f(x) = g(x)
sin(x) * cos(x) = 2cos(x)
Step 2: Solve for x.
We can simplify the equation by dividing both sides by cos(x):
sin(x) = 2
However, cos(x) cannot be zero for the equation to be valid. Therefore, we have two separate cases to consider:
Case 1: x = π + 2kπ, where k is an integer
In this case, sin(x) = 0 and the original equation becomes:
0 = 2cos(x)
This equation has no solutions, so there are no points of intersection in this case.
Case 2: x = π/2 + 2kπ, where k is an integer
In this case, sin(x) = 1 and the original equation becomes:
1 = 2cos(x)
cos(x) = 1/2
This equation has two solutions within the given interval (0 ≤ x < 2π):
x = π/3
x = 5π/3
Step 3: Express the solutions in terms of π.
The two points of intersection are:
(π/3, 1)
(5π/3, 1)