A point starts at the location (−4,0) and moves CCW along a circle centered at (0,0) at a constant angular speed of 2 radians per second. Let t represent the number of seconds since the point has swept out since it started moving. Draw a diagram of this to make sure you understand the context!
Suppose the point has traveled for 0.25 seconds (t=0.25). How many radians would need to be swept out from the 3-o'clock position [or from (4,0)] to get to the point's current position?
3.641593 radians
Write an expression in terms of t to represent how many radians would need to be swept out from the 3-o'clock position to get to the point's current position.
Pi+(t×2)
Write a function g that determine's the point's x-coordinate in terms t.
g(t) = -4cos(Pi+(t×2))
Sure, here's the diagram you requested:
```
* (0,4)
/
/
/
/
/
/
* (-4,0)
```
As for the expression representing how many radians would need to be swept out from the 3-o'clock position to get to the point's current position, it would be π + (t × 2).
Now, for the function g that determines the point's x-coordinate in terms of t:
g(t) = -4 × sin(2t)
The x-coordinate of the point in terms of t can be determined using the trigonometric relationship between the angle swept out and the x-coordinate of a point on the circle.
The x-coordinate can be expressed as: g(t) = r * cos(angle), where r is the radius of the circle and angle is the angle swept out by the point.
In this case, the radius of the circle is 4 units (the distance from the center (0,0) to the initial point (-4,0)).
So, the expression for the x-coordinate in terms of t would be: g(t) = 4 * cos(angle).
Now, we need to find the value of angle in terms of t. Considering that the point has a constant angular speed of 2 radians per second, we can express the angle swept out as: angle = (2 * t) - π.
Therefore, the function g(t) would be: g(t) = 4 * cos((2 * t) - π).
The x-coordinate of the point can be determined using the equation of a circle: x = r * cos(angle). In this case, the radius of the circle is the distance from the center (0,0) to the initial position (-4,0), which is 4 units.
Since the point moves counterclockwise at a constant angular speed of 2 radians per second, the angle swept out at a given time t can be represented as 2t radians.
Using the equation x = 4 * cos(2t), we can define the function g(t) to determine the point's x-coordinate in terms of time:
g(t) = 4 * cos(2t)