Problem 31 - Algebra and Trigonometry

Question

Suppose that the terminal point determined by \(t\) is the point \(\left(\frac{3}{5}, \frac{4}{5}\right)\) on the unit circle. Find the terminal point determined by each of the following. \(\begin{array}{ll}{\text { (a) } \pi-t} & {\text { (b) }-t} \\ {\text { (c) } \pi+t} & {\text { (d) } 2 \pi+t}\end{array}\)

Step-by-Step Solution

Answer

(a) \( \left(\frac{3}{5}, -\frac{4}{5}\right) \), (b) \( \left(\frac{3}{5}, -\frac{4}{5}\right) \), (c) \( \left(-\frac{3}{5}, -\frac{4}{5}\right) \), (d) \( \left(\frac{3}{5}, \frac{4}{5}\right) \).

Step 1: Understanding the Unit Circle Point

The given point \( \left(\frac{3}{5}, \frac{4}{5}\right) \) is on the unit circle, meaning it is determined by angle \( t \). This point forms a right triangle with the origin and has coordinates \( x = \frac{3}{5} \) and \( y = \frac{4}{5} \).

Step 2: Terminal Point for \( \pi-t \)

For \( \pi-t \), we reflect the point across the x-axis. The x-coordinate remains the same, while the y-coordinate changes sign. So, the point is \( \left(\frac{3}{5}, -\frac{4}{5}\right) \).

Step 3: Terminal Point for \( -t \)

For the angle \( -t \), reflect the point across the x-axis, resulting again in the point \( \left(\frac{3}{5}, -\frac{4}{5}\right) \).

Step 4: Terminal Point for \( \pi+t \)

For \( \pi+t \), reflect the point across both the x and y-axis which means reversing both signs of the coordinates: \( \left(-\frac{3}{5}, -\frac{4}{5}\right) \).

Step 5: Terminal Point for \( 2\pi+t \)

For \( 2\pi+t \), complete a full rotation around the circle, bringing the point back to its original position: \( \left(\frac{3}{5}, \frac{4}{5}\right) \).

Key Concepts

Unit Circle

The unit circle is a foundational concept in trigonometry. It is a circle with a radius of 1, centered at the origin of the coordinate plane. This means every point on the unit circle satisfies the equation:

• \[ x^2 + y^2 = 1 \]

Points on the unit circle correspond to angles, measured in radians, from the positive x-axis. The coordinates of each point are

• \( x = \cos(t) \)
• \( y = \sin(t) \)

where \( t \) is the angle in radians. For example, the point \( \left(\frac{3}{5}, \frac{4}{5}\right) \) signifies an angle \( t \) such that \( \cos(t) = \frac{3}{5} \) and \( \sin(t) = \frac{4}{5} \). This is crucial when analyzing transformations and reflections of points on the unit circle.

Terminal Point

The terminal point is the point on the unit circle that corresponds to a given angle \( t \). This is key in understanding how angles relate to positions on the circle. For instance, when we say a point is determined by \( t \), we're saying \( x = \cos(t) \) and \( y = \sin(t) \) describe that point's coordinates. This helps when considering angle changes that affect these coordinates:

• For \( t = 0 \), the terminal point is \((1, 0)\).
• For \( t = \frac{\pi}{2} \), the terminal point moves to \((0, 1)\).

Manipulating \( t \) allows different positions and points to be explored via trigonometric functions, solidifying the role of angles in coordinate positioning.

Angle Reflection

Angle reflection involves changing the position of the terminal point to a new location on the unit circle. This is done by reflecting across one or more axes:

• Reflection across the x-axis involves changing the sign of the y-coordinate. Thus, \(\left(\frac{3}{5}, \frac{4}{5}\right)\) becomes \(\left(\frac{3}{5}, -\frac{4}{5}\right)\).
• Reflection across the y-axis sees a change in the sign of the x-coordinate, such as \(\left(\frac{3}{5}, \frac{4}{5}\right)\) becoming \(\left(-\frac{3}{5}, \frac{4}{5}\right)\).
• Reflection across both axes, reverses both coordinates' signs, resulting in a new point like \(\left(-\frac{3}{5}, -\frac{4}{5}\right)\).

These reflections help in visualizing how different angles can be achieved from a given point by altering its orientation.

Coordinate Transformation

Coordinate transformation is the changing of a point's position through systematic operations like rotations or translations. In the context of the unit circle, it often explores rotating angles to see the effect on coordinates:

• For a rotation of \( \pi \), the point is inverted to \(\left(-\frac{3}{5}, -\frac{4}{5}\right)\).
• With a rotation of \( 2\pi \), there is a full circle and the point returns to its initial spot \(\left(\frac{3}{5}, \frac{4}{5}\right)\).

These transformations are significant in understanding periodicity in trigonometric functions and ensuring conceptual clarity on how rotations affect any point on the unit circle.