Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-8: Let \(P(t)\) be the point on the unit circle \(U\) that corresponds to \(t\). If \(P(t)\) has the given rectangular coordinates, find (a) \(P(t+\pi)\) (b) \(P(t-\pi)\) (c) \(P(-t)\) (d) \(P(-t-\pi)\) $$ \left(\frac{7}{25},-\frac{24}{25}\right) $$

Step-by-Step Solution

Answer

(a) \((-\frac{7}{25}, \frac{24}{25})\), (b) \((-\frac{7}{25}, \frac{24}{25})\), (c) \((\frac{7}{25}, \frac{24}{25})\), (d) \((-\frac{7}{25}, -\frac{24}{25})\).

Step 1: Understanding the Problem

We need to find four different points on the unit circle related to the given point \(P(t) = \left(\frac{7}{25},-\frac{24}{25}\right)\). The points to be found are \(P(t+\pi)\), \(P(t-\pi)\), \(P(-t)\), and \(P(-t-\pi)\). These points are transformations based on certain angular shifts on the unit circle.

Step 2: Find \(P(t+\pi)\) using symmetry properties

On the unit circle, adding \(\pi\) to the angle corresponds to finding the opposite point, as it rotates 180 degrees. Given \(P(t) = (x, y) = \left(\frac{7}{25}, -\frac{24}{25}\right)\), the opposite point is \((-x, -y)\). So, \(P(t+\pi) = \left(-\frac{7}{25}, \frac{24}{25}\right)\).

Step 3: Calculate \(P(t-\pi)\) using symmetry properties

Similarly, subtracting \(\pi\) is the same as adding \(\pi\), due to the properties of the circle, thus \(P(t-\pi) = P(t+\pi) = \left(-\frac{7}{25}, \frac{24}{25}\right)\).

Step 4: Determine \(P(-t)\) with reflection over the x-axis

The \(-t\) reflects the point across the x-axis. For \(P(t) = (x, y) = \left(\frac{7}{25}, -\frac{24}{25}\right)\), reflection leads to \(P(-t) = (x, -y) = \left(\frac{7}{25}, \frac{24}{25}\right)\).

Step 5: Find \(P(-t-\pi)\) by another reflection and rotation

Start with \(P(-t) = \left(\frac{7}{25}, \frac{24}{25}\right)\). Adding (or subtracting) \(\pi\) to this reflects it again to the opposite, resulting in \(P(-t-\pi) = \left(-\frac{7}{25}, -\frac{24}{25}\right)\).

Key Concepts

Trigonometric Transformations

Trigonometric transformations on the unit circle tie together angles, coordinates, and their transformations.
For instance, when transforming an angle by adding or subtracting \(\pi\), it corresponds to shifting the point to the opposite side of the circle.
This operation changes the sign of both \(x\) and \(y\) coordinates of the original point.

• Adding \(\pi\) or subtracting \(\pi\) to an angle, as seen in \(t+\pi\) and \(t-\pi\), results in a 180-degree rotation, which flips the signs of the coordinates.
• Reflecting with \(-t\) means altering the angle to its negative, swapping the sign of the \(y\)-coordinate, reflecting across the x-axis.
• The combination transformation \(-t-\pi\) involves reflecting and then rotating, transforming the coordinates to their opposite values both in \(x\) and \(y\).

These transformations are crucial in understanding periodic patterns in trigonometry and the behavior of trigonometric functions.

Symmetry Properties

Symmetry properties are fundamental for comprehending movements on the unit circle.
When a point on the unit circle is transformed by adding \(\pi\), it shifts to the directly opposite side, making it symmetric.

• This symmetry means if a point \(P(t)\) is \((x, y)\), then its symmetric point \(P(t+\pi)\) is \((-x, -y)\).
• Similarly, subtracting \(\pi\), or \(t-\pi\), confers the same symmetric result as \(t+\pi\).

These symmetry properties are valuable because they easily allow us to predict the resulting coordinates after these transformations without mechanical calculations.

Angle Reflection

Angle reflection on the unit circle highlights the role of reflection over axes in transforming angles.
For a point \(P(t) = \left(\frac{7}{25}, -\frac{24}{25}\right)\), reflecting it over the x-axis (i.e., finding \(P(-t)\)) inverts the \(y\)-coordinate while keeping the \(x\)-coordinate constant.

• The new point appears as \((x, -y)\).
• Furthermore, applying reflection followed by a half-turn \(-t-\pi\) brings the entire point to its opposite with both coordinates inverted \((-x, -y)\).

This is illustrative of how reflections modify points in a simple yet definitive manner, helping one to visualize transformations intuitively.

Coordinate Geometry

Coordinate geometry on the unit circle is a powerful tool to visualize the transformations of points.
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.

• The coordinates of any point \(P(t)\) on this circle are \((x, y)\), where \(x = \cos(t)\) and \(y = \sin(t)\).
• Given a coordinate such as \(\left(\frac{7}{25}, -\frac{24}{25}\right)\), transformations are calculated directly through simple arithmetic involving \(x\) and \(y\).

This makes the unit circle an especially efficient way to apply trigonometric transformations, leveraging the consistency of geometric properties to simplify complex computations.