Problem 25 - Algebra 2 and Trigonometry

Question

Under a reflection in the \(y\) -axis, the image of \(A(x, y)\) is \(A^{\prime}(-x, y)\) . The measure of \(\angle R O P=\theta\) and \(P(\cos \theta, \sin \theta)\) is a point on the terminal side of \(\angle R O P .\) Let \(P^{\prime}\) be the image of \(P\) and \(R^{\prime}\) be the image of \(R\) under a reflection in the \(y\) -axis. a. What are the coordinates of \(P^{\prime} ?\) b. Express the measure of \(\angle R^{\prime} O P^{\prime}\) in terms of \(\theta\) c. Express the measure of \(\angle R O P^{\prime}\) in terms of \(\theta\)

Step-by-Step Solution

Answer

a. \((-\cos \theta, \sin \theta)\), b. \(\pi - \theta\), c. \(-\theta\) or \(\theta\).

Step 1: Reflect Point P across the y-axis

When a point \( P(x, y) \) is reflected across the \( y \)-axis, its \( x \)-coordinate changes sign while the \( y \)-coordinate remains unchanged. Given \( P(\cos \theta, \sin \theta) \), the reflection \( P' \) will be at \( (-\cos \theta, \sin \theta) \). Thus, the coordinates of \( P' \) are \( (-\cos \theta, \sin \theta) \).

Step 2: Reflect point R across the y-axis

Just as with point \( P \), if \( R(x_r, y_r) \) is the initial position of \( R \), then its image under a reflection over the \( y \)-axis will be \( R'(-x_r, y_r) \). However, since the exact coordinates of \( R \) are not specified, we focus on the effect of the reflection on the angle positions.

Step 3: Measure angle \( \angle R' O P' \)

The angle \( \angle R' O P' \) after reflection can be considered by noting the change in \( P \). Since \( P \) initially at \( (\cos \theta, \sin \theta) \) is reflected to \( (-\cos \theta, \sin \theta) \), the angle it makes, relative to the positive \( x \)-axis after reflection, is \( \pi - \theta \). Therefore, \( \angle R' O P' = \pi - \theta \).

Step 4: Measure angle \( \angle R O P' \)

The angle \( \angle R O P' \) involves the initial point \( R \) and the reflected point \( P' \). Since \( P'(-\cos \theta, \sin \theta) \) again lies in the second quadrant, the angle \( \angle R O P' \) from the \( x \)-axis is \( \pi - \theta \), due to symmetry across the \( y \)-axis. However, if \( R \) remains unaffected, \( \angle R O P' = \theta - 2\theta = -\theta \); or if considering counterclockwise angles only, \( \theta \).

Key Concepts

Reflection

A reflection is a transformation that 'flips' a point or object over a specific line. In the context of the problem, the reflection of a point across the y-axis involves changing the sign of its x-coordinate, while the y-coordinate stays the same.
This means for a point \( A(x, y) \), its reflected image will be \( A'(-x, y) \).
This fundamental transformation is used to find the image of the point \( P(\cos \theta, \sin \theta) \) across the y-axis, resulting in \( P'(-\cos \theta, \sin \theta) \). This technique forms the basis for understanding more complex geometric transformations, especially when dealing with coordinates.

• Reflection involves a 'mirror image' effect.
• Important for understanding symmetry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to explore geometric properties by connecting them with algebraic equations.
In this exercise, the point \( P(\cos \theta, \sin \theta) \) lies on the unit circle, combining the concepts of trigonometry and geometry. The coordinates \( (\cos \theta, \sin \theta) \) are derived from the basic trigonometric functions, cosine and sine, which represent x and y coordinates, respectively, on the unit circle.
Reflections are important transformations in coordinate geometry, allowing for evaluation of symmetry properties, especially across axes.

• Coordinate geometry links algebra with geometric concepts.
• It helps in visualizing geometric transformations like reflections.

Angle Measurement

Angles are measured in degrees or radians, providing a way to express the orientation or rotation of a figure in geometry. This exercise uses radians, with the angle \( \theta \) specifying the point \( P(\cos \theta, \sin \theta) \) on the circle relative to the positive x-axis.
After reflection performing the reflection, angles are adjusted accordingly. A reflection over the y-axis changes the angle from \( \theta \) to \( \pi - \theta \) because the point flips to the opposite side of the x-axis while maintaining the same y-coordinate.
This reflects the symmetry of angles through the understanding of their measurement in the coordinate plane.
While degrees might be more common in everyday use, radians are often preferred in mathematics due to their natural connection to the unit circle.

• Angle measurement is crucial for describing orientation.
• Understanding radians is key for advanced geometry and trigonometry.

Quadrants

The Cartesian coordinate system is divided into four quadrants, each representing a distinct combination of positive and negative values for x and y coordinates.
This exercise involves the use of the second quadrant, where coordinates \((-\cos \theta, \sin \theta)\) lie. Here, the x-values are negative, and y-values are positive. Reflections often lead to changes in quadrant, altering the angle representation. Each quadrant affects the trigonometric signs:

• Quadrant I: Positive x and y values
• Quadrant II: Negative x, Positive y values
• Quadrant III: Negative x and y values
• Quadrant IV: Positive x, Negative y values

Understanding quadrants is crucial because they affect the sign and value of angles and coordinates, which is central to correct evaluation after any transformation like reflection.