Finding Rotation Angle: Point P To P' Transformation

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Hey everyone! Today, we're diving into a cool math problem: figuring out the rotation angle when a point is spun around the origin. We'll be using some geometry and trigonometry to crack the code. So, let's get started, shall we?

Understanding the Problem: Rotation in the Coordinate Plane

Okay, guys, the core concept here is rotation. Imagine you've got a point, let's call it P, sitting on a coordinate plane. Now, we're going to spin this point around another point, which we'll call the center of rotation. In our case, the center is the origin, O (0, 0). The point P starts at coordinates (-5, -6), and after the rotation, it magically transforms into a new point, P' with coordinates (-6, 5). Our mission, if we choose to accept it, is to determine the angle of this rotation, which is represented by a degrees. Essentially, we want to know how many degrees we spun P to get to P'. This involves understanding how points change when rotated around the origin, and then using that understanding to calculate the angle.

To make things easier, picture this: each point (P and P') has a distance from the origin. This distance doesn't change during rotation. The angle is the amount that point has to be rotated. The process involves some simple trigonometry, which makes it understandable. The key to solving this problem lies in understanding that rotations preserve distances and involve trigonometric functions. This is not the complicated calculus you might be thinking of, but pretty simple and understandable. This means that if we can identify how the x and y coordinates change from P to P', we can determine the angle of rotation. We can use the information given, and employ our prior knowledge of trigonometric functions, such as sine, cosine, and tangent. Also, we will use the understanding of how points change when they are rotated around the origin and then calculate the angle of rotation using these formulas. Remember, the rotation is centered at the origin, and we're looking at a single point in the plane. It is also important to consider the quadrant that the point is in, before and after the rotation. This will help to determine the sign of the angle. Let's get cracking!

Visualizing the Points and the Rotation

First things first, it's super helpful to visualize what's going on. Picture the coordinate plane, with the x-axis going horizontally and the y-axis going vertically. Mark the point P at (-5, -6). This puts it in the third quadrant. Then mark the point P' at (-6, 5). This puts P' in the second quadrant. This visual representation is very useful. It is beneficial to draw a quick sketch to get a feel for the situation. It lets you get an idea of the rotation. The distance from the origin to P is the same as the distance from the origin to P'. That distance, which is the radius of the circle that P and P' lie on, remains the same throughout the rotation. Think of it like a clock hand spinning around the center. The hand's length stays the same, but the angle changes as it moves. The distance from the origin to the point stays constant. When you look at the graph, it should immediately give you an idea of the general direction of the rotation. It will also help you check if your final answer makes sense. Visualizing the rotation helps in understanding the relationship between the original point, the rotated point, and the angle. This process makes it easier to keep track of the signs of the coordinates and to understand how they change during the rotation. By visualizing the points, we can understand the direction of rotation (clockwise or counterclockwise) and estimate the angle. This visualization is essential for understanding the problem before jumping into the calculations.

Using Trigonometry to Find the Angle

Alright, time to get our math on! The standard way to approach this involves using trigonometric functions, specifically the arctangent function. The arctangent function helps us find the angle, given the ratio of the opposite side to the adjacent side in a right triangle. Since our rotation is about the origin, we can think of the x and y coordinates of the points as sides of a right triangle. The distance from the origin to the point becomes the hypotenuse. We can use the original coordinates of P (-5, -6) and P' (-6, 5) to help calculate the angle of rotation. We'll denote the angle from the positive x-axis to the line segment OP as θ1 and the angle from the positive x-axis to the line segment OP' as θ2. The angle of rotation a is the difference between θ2 and θ1 (a = θ2 - θ1).

Let's calculate θ1 first. We have the coordinates of P (-5, -6). Therefore, x = -5 and y = -6. The tangent of θ1, tan(θ1), is y/x, which is -6/-5 = 6/5. Then, θ1 = arctan(6/5). Because P is in the third quadrant, we need to add 180° to get the correct angle (arctan(6/5) + 180°). Now, let’s calculate θ2. For P' (-6, 5), x = -6 and y = 5. So, tan(θ2) = 5/-6 = -5/6. Therefore, θ2 = arctan(-5/6). Because P' is in the second quadrant, we need to add 180° to get the correct angle (arctan(-5/6) + 180°). Finally, to find the rotation angle a, we'll subtract θ1 from θ2. This involves using the arctangent function to find the angles relative to the x-axis, and then computing the difference. This process may sound complex, but with the help of a calculator, it should be simple. The key is to correctly identify the quadrant and adjust the angle accordingly. Make sure to double-check that your calculator is in degree mode. The angles can then be plugged into the formula to find the rotation angle a. So, with some careful calculations, and keeping track of our quadrants, we should be able to get the solution.

Calculating the Angle Step-by-Step

Okay, guys, let’s get into the step-by-step process to calculate the angle a. Follow along closely, as this is where the magic happens!

  1. Find θ1 (Angle of P):

    • Using the coordinates of P (-5, -6), tan(θ1) = -6 / -5 = 6/5. Therefore, θ1 = arctan(6/5). Since P is in the third quadrant, we add 180° to get the actual angle. So, θ1 ≈ arctan(1.2) + 180° ≈ 50.19° + 180° ≈ 230.19°.

  2. Find θ2 (Angle of P'):

    • Using the coordinates of P' (-6, 5), tan(θ2) = 5 / -6 = -5/6. Therefore, θ2 = arctan(-5/6). Since P' is in the second quadrant, we add 180° to get the actual angle. So, θ2 ≈ arctan(-0.833) + 180° ≈ -39.81° + 180° ≈ 140.19°.

  3. Calculate the Rotation Angle a:

    • a = θ2 - θ1. Thus, a ≈ 140.19° - 230.19° ≈ -90°. Because the angle is negative, we add 360° to get the equivalent positive angle, which is 270°. We see that a rotation of 270 degrees is the same as a rotation of -90 degrees. This step is the crucial calculation where we determine the angle of rotation by subtracting the original angle from the rotated angle. The negative sign means the rotation is clockwise. It is vital to pay close attention to the direction of rotation. The direction of rotation can be clockwise or counter-clockwise. For this problem, we've found that rotating the point P by 270 degrees counterclockwise (or -90 degrees clockwise) will result in the point P'. When working with angles in trigonometry, it is essential to be mindful of the quadrants in which the points lie, as the quadrant influences the angle's value. The final result represents the total angle of rotation necessary to move P to P'.

Conclusion: The Final Answer

So, after all that work, we've found our answer, guys! The angle of rotation a is approximately 270 degrees (or -90 degrees). This means that to get from point P (-5, -6) to point P' (-6, 5), we rotated the point 270 degrees counterclockwise (or 90 degrees clockwise) around the origin. That's a wrap! I hope this helps you understand how rotations work in the coordinate plane. Remember the key steps: visualize the points, use trigonometry to find the angles, and keep track of the quadrants! Don’t worry if it seems tricky at first; with practice, it’ll become second nature. Understanding how to handle rotations is a core skill in math and can be applied in various real-world scenarios, from computer graphics to engineering. Great job, everyone. Keep practicing, and you'll be math whizzes in no time! Keep exploring the world of math, and you will understand more complex concepts! Until next time!