Consider a circle of radius 5 and a point P rotating around it, as shown in the figure to the left, below.

The angle(theta) , in radians, is given as a function of time t, by the graph in the second figure.

What is the value of when t=0.5?
Find the coordinates of P when t=0.5

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To find the value of when t = 0.5, we need to refer to the graph that shows the angle theta as a function of time. Since the graph is not provided, I'm unable to compute the value of theta at t = 0.5.

However, once we know the value of theta, we can find the coordinates of point P on the circle using trigonometric functions.

Let's assume that the initial position of point P is at (5, 0) on the coordinate plane (assuming the center of the circle is at the origin).

If we know the angle theta, we can find the x-coordinate of point P using the trigonometric function cosine:
x = r * cos(theta),

where r is the radius of the circle (which is 5 in this case).

Similarly, we can find the y-coordinate of point P using the trigonometric function sine:
y = r * sin(theta).

Therefore, to find the coordinates of point P when t = 0.5, we need to know the value of theta at t = 0.5.

To find the value of θ when t = 0.5, we need to analyze the given graph of θ as a function of time t.

1. Look at the x-axis of the graph to find the value of t = 0.5.
2. Then, locate the corresponding point on the graph that corresponds to t = 0.5.
3. Read the y-coordinate of that point, which represents the value of θ when t = 0.5.

Now, to find the coordinates of point P when t = 0.5, we can use the value of θ obtained in the previous step. Since P is rotating around a circle of radius 5, we can use trigonometry to determine the x and y coordinates.

1. Apply the formula x = r * cos(θ) and substitute the value of θ obtained earlier, along with the radius r = 5, to find the x-coordinate of P.
2. Similarly, apply the formula y = r * sin(θ) and substitute the value of θ to find the y-coordinate of P.

By following these steps, you can calculate both the value of θ and the coordinates of point P when t = 0.5.