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Let's break down the problem using the given point \((-2, 2)\) to find the trigonometric properties of \(\theta\) and solve the subparts.

### Part a: Find \( P(\theta) \)
The point \((-2, 2)\) is on the terminal side of angle \(\theta\). This point lies in the second quadrant because \(x\) is negative and \(y\) is positive.

To find \(P(\theta)\), we need to calculate the sine and cosine of \(\theta\).

1. **Calculate the radius \(r\)**:
   \[
   r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
   \]
   
2. **Find \(\cos(\theta)\)**:
   \[
   \cos(\theta) = \frac{x}{r} = \frac{-2}{2\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}
   \]
   
3. **Find \(\sin(\theta)\)**:
   \[
   \sin(\theta) = \frac{y}{r} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
   \]

Thus, \(P(\theta) = (\cos(\theta), \sin(\theta)) = \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\).

### Part b: Find \(P(2\pi - \theta)\)
We now need to find the coordinates of the angle \(2\pi - \theta\), which is the reference angle in the fourth quadrant. In the fourth quadrant, the sine is negative, and the cosine remains the same as it is in the second quadrant.

1. **Cosine remains the same**:
   \[
   \cos(2\pi - \theta) = \cos(\theta) = -\frac{\sqrt{2}}{2}
   \]
   
2. **Sine changes sign**:
   \[
   \sin(2\pi - \theta) = -\sin(\theta) = -\frac{\sqrt{2}}{2}
   \]

Thus, \(P(2\pi - \theta) = \left(\cos(2\pi - \theta), \sin(2\pi - \theta)\right) = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\).

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Would you like more details on any of these steps or have any further questions? Here are five related questions you can explore:

1. How would the coordinates change if the point were in the third quadrant?
2. How do you derive the values of sine and cosine from any given point?
3. What is the significance of the angle \(2\pi - \theta\) in trigonometry?
4. How do the signs of sine and cosine differ across the four quadrants?
5. Can you use the same method to find \(P(\pi + \theta)\)?

**Tip**: When working with trigonometric functions, always pay attention to the signs of sine and cosine in different quadrants!

Given that the terminal side of θ contains the point (-2, 2), find P(θ) and P(2π - θ).

Learn how to find the trigonometric function P(θ) when the terminal side of θ passes through the point (-2, 2). This problem explores cosine and sine calculations, properties of angles in different quadrants, and the unit circle. Suitable for students in Grades 10-12.

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