https://file.aiquickdraw.com/mathbot/file/1726615254330nim9yn1w.jpg

Let's break down the question from the image and solve it.

We have:
1. A circle of radius 2 centered at the origin, with a point \( P \) rotating around it.
2. The graph of \( \theta(t) \), the angle in radians as a function of time \( t \), is shown in the second figure.

The two parts of the question are:

**(a)** What is the value of \( \theta \) when \( t = 1 \)?

From the graph of \( \theta(t) \) on the right, when \( t = 1 \), it looks like \( \theta \approx \frac{\pi}{2} \) or \( 1.57 \) radians.

So,
\[
\theta(1) = \frac{\pi}{2}
\]

**(b)** Find the coordinates of \( P \) when \( t = 1 \).

Since \( P \) is on the circle, its coordinates are given by:
\[
(x, y) = (2 \cos \theta, 2 \sin \theta)
\]
At \( t = 1 \), we have \( \theta = \frac{\pi}{2} \). Therefore,
\[
x = 2 \cos \left( \frac{\pi}{2} \right) = 2(0) = 0
\]
\[
y = 2 \sin \left( \frac{\pi}{2} \right) = 2(1) = 2
\]
Thus, the coordinates of \( P \) are \( (0, 2) \).

Let me know if you have any questions or need further clarification!

Here are 5 related questions:
1. What are the coordinates of \( P \) when \( \theta = \pi \)?
2. How do the coordinates of \( P \) change as \( \theta \) increases?
3. What is the angular velocity \( d\theta/dt \) at \( t = 1 \) based on the graph?
4. How would the coordinates change if the radius of the circle were 3 instead of 2?
5. What happens to the point \( P \) when \( \theta = 0 \)?

**Tip:** When working with parametric equations involving circles, always remember the basic formulas \( x = r \cos \theta \) and \( y = r \sin \theta \).

Consider a circle of radius 2 and a point P rotating around it. The angle θ, in radians, is given as a function of time t by the graph. (a) What is the value of θ when t = 1? (b) Find the coordinates of P when t = 1.

Learn how to find the angle θ and the coordinates of a point P rotating around a circle of radius 2. Includes a step-by-step solution using trigonometric functions. Suitable for high school students studying geometry and trigonometry.

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