Problem 32 - Algebra and Trigonometry

Question

Suppose that the terminal point determined by \(t\) is the point \(\left(\frac{3}{4}, \sqrt{7} / 4\right)\) on the unit circle. Find the terminal point deter- mined by each of the following. \(\begin{array}{ll}{\text { (a) }-t} & {\text { (b) } 4 \pi+t} \\ {\text { (c) } \pi-t} & {\text { (d) } t-\pi}\end{array}\)

Step-by-Step Solution

Answer

(a) \(\left(\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\); (b) \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\); (c) \(\left(-\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\); (d) \(\left(-\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\).

Step 1: Understanding the Problem

We are given a terminal point \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\) on the unit circle determined by angle \(t\). We need to find the terminal points for several transformations of \(t\): \(-t,\ 4\pi+t,\ \pi-t,\ \text{and } t-\pi\). Since the unit circle has a radius of 1, reflecting or shifting angles will result in points that also lie on the unit circle.

Step 1: Finding the Point for \(-t\)

The terminal point for \(-t\) is found by reflecting the original point across the x-axis. Therefore, the x-coordinate remains the same, and the y-coordinate's sign is inverted. The terminal point for \(-t\) is \(\left(\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\).

Step 2: Finding the Point for \(4\pi + t\)

Adding \(4\pi\) to \(t\) results in a full revolution around the circle and returns to the original point. Hence, the terminal point for \(4\pi + t\) is \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\).

Step 3: Finding the Point for \(\pi - t\)

The transformation \(\pi - t\) reflects the original point across the y-axis. The x-coordinate changes sign while the y-coordinate remains the same. Thus, the terminal point for \(\pi - t\) is \(\left(-\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\).

Step 4: Finding the Point for \(t - \pi\)

Subtracting \(\pi\) from \(t\) rotates the original point halfway around the circle. Therefore, both coordinates change signs. The terminal point for \(t - \pi\) is \(\left(-\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\).

Key Concepts

Angles and Rotation

Understanding angles and rotation is key when studying the unit circle. An angle, in this context, is measured in radians, which is a way of measuring angles based on the radius of a circle. One complete rotation around a circle is equivalent to an angle of \(2\pi\) radians. This understanding allows us to interpret transformations on a circle. When a transformation involves adding \(4\pi\) to an angle, it means performing a complete rotation and returning to the same point on the circle.

Rotational transformations are usually expressed in radians. Simple transformations such as adding multiples of \(\pi\) rotate point positions around a circle. These transformations result in shifts to different quadrants, which create new terminal points. By comprehending angles and their rotations, you become able to predict how points move around the circle with each transformation. Understanding this concept helps you grasp the relationship between an angle and its corresponding point on the unit circle.

Reflection Across Axes

Reflection is a geometrical transformation that flips points across a line (axis). In the unit circle context, reflecting a point modifies one of its coordinates:

• Reflection across the x-axis reverses the sign of the y-coordinate, leaving the x-coordinate unchanged.
• Reflection across the y-axis switches the sign of the x-coordinate, leaving the y-coordinate the same.

These reflections are crucial in finding terminal points for transformations like \( -t \) and \( \pi - t \). For example, in the case of \( -t \), the point \( \left( \frac{3}{4}, \frac{\sqrt{7}}{4} \right) \) becomes \( \left( \frac{3}{4}, -\frac{\sqrt{7}}{4} \right) \) after reflecting across the x-axis. Reflections help us visualize changes without needing to perform physical rotations on the circle.

Coordinate Transformation

Coordinate transformation refers to the process of converting between coordinate systems or altering the coordinates of a point. In the context of the unit circle, transformations often involve changing sign or adding specific values, all with respect to angle changes.

The transformations given in the exercise essentially modify the angles, resulting in different terminal points through shifts and reflections. For example:

• Equation modifications like \(t - \pi\) rotate a point 180 degrees around the circle, flipping both the x and the y coordinates.
• Such transformations help us understand how angles and positions are interconnected by visualizing the circle as a system where coordinates change as angles are adjusted.

Understanding coordinate transformations aids in solving complex rotation and position exercises on the unit circle.

Terminal Points on a Circle

Terminal points on a circle are the final positions a point reaches on the circle after traveling through specific angles. In the unit circle, these points have coordinates that are derived from the cosine and sine of the angle in question.

Every angle corresponds to a specific point, and transformations of these angles result in shifts around the circle to different coordinates. For instance:

• If the terminal point at angle \(t\) is \( \left( \frac{3}{4}, \frac{\sqrt{7}}{4} \right) \), transformations like \(\pi - t\) or \(t - \pi\) will move the point to positions such as \( \left(-\frac{3}{4}, \frac{\sqrt{7}}{4}\right) \) or \( \left(-\frac{3}{4}, -\frac{\sqrt{7}}{4}\right) \), respectively.
• These positions help visualize how angle adjustments change locations around the circle, offering insight into trigonometric functions and their graphical representations on the unit circle.

Grasping the concept of terminal points is integral for solving related problems and understanding the dynamic nature of angles and their corresponding locations.