Problem 34 - 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: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system. Any point on the unit circle represents an angle's cosine and sine values, where the x-coordinate is the cosine and the y-coordinate is the sine of the angle.

Step 2: Given Terminal Point Interpretation

The point given is \( \left( \frac{3}{4}, \frac{\sqrt{7}}{4} \right) \). This implies that the cosine of angle \( t \) is \( \frac{3}{4} \) and the sine of angle \( t \) is \( \frac{\sqrt{7}}{4} \). Use these to interpret angle reflections and translations.

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

The terminal point for \(-t\) is a reflection over the x-axis. This means the x-coordinate remains \( \frac{3}{4} \), but the y-coordinate becomes negative. So, the terminal point is \( \left( \frac{3}{4}, -\frac{\sqrt{7}}{4} \right) \).

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

Adding \(4\pi\) corresponds to two full rotations on the unit circle, which does not alter the terminal point. Thus, the terminal point remains \( \left( \frac{3}{4}, \frac{\sqrt{7}}{4} \right) \).

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

The terminal point for \(\pi - t\) is a reflection over the y-axis. Thus, the x-coordinate changes its sign but the y-coordinate remains the same. Hence, the terminal point is \( \left( -\frac{3}{4}, \frac{\sqrt{7}}{4} \right) \).

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

The terminal point for \(t - \pi\) involves rotating the point 180 degrees or a reflection over the origin, which changes the signs of both coordinates. Therefore, the terminal point is \( \left( -\frac{3}{4}, -\frac{\sqrt{7}}{4} \right) \).

Key Concepts

Terminal Points

In the context of a unit circle, a terminal point is the specific point where an angle's terminal side intersects the circle. Imagine the circle drawn on a coordinate system with the center at the origin \(0,0\). For any given angle, measured from the positive x-axis, its terminal side rotates until it points at a unique location on the circle's edge.

This point is crucial as it provides direct values for trigonometric functions. Specifically, the x-coordinate of the terminal point represents the cosine, while the y-coordinate provides the sine value for that angle.

• Terminal points help translate angles into precise coordinates on a unit circle.
• Each can be manipulated by understanding the fundamental properties and symmetries of the circle.

Reflections on the Unit Circle

Reflections on the unit circle involve flipping points over an axis such as the x-axis or y-axis. These reflections are pivotal in modifying angles to find equivalent positions on the circle.

For instance, if you reflect a terminal point over the x-axis, you flip the y-value while maintaining the x-value. This process turns cosine and sine values negative or positive depending on their original positions.

• Reflection over the x-axis: changes the sign of the sine value but leaves the cosine unchanged.
• Reflection over the y-axis: changes the sign of the cosine value but keeps the sine unchanged.
• Reflection over the origin (180-degree rotation): alters signs of both coordinates.

Understanding reflections allows for finding terminal points for negative angles or angle complements, which are often used in trigonometry problems.

Angle Transformations

Angle transformations involve modifying angles using operations like addition, subtraction, or rotation. On the unit circle, these transformations can result in different but equivalent points.

Adding \(2\pi\) or any multiple of it to an angle represents full rotations, leading back to the same terminal point on the circle. For instance:

• Adding \(4\pi\) to an angle results in two full rotations, so the terminal point remains unchanged.
• Transformations like \(-t\) or \(\pi - t\) utilize reflections, altering positions, as discussed in reflections.
• Similarly, transformations such as \( - \pi\) cause a 180-degree rotation offering a completely opposite point on the circle.

Utilizing these transformations helps solve trigonometric problems by simplifying angles to more manageable values.

Cosine and Sine Values

Cosine and sine are fundamental trigonometric functions that relate directly to coordinates on the unit circle.

The x-coordinate of a point on the unit circle gives the cosine of the corresponding angle, while the y-coordinate gives the sine. With coordinates such as \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\), you directly know:

• The cosine value is \frac{3}{4}\.
• The sine value is \frac{\sqrt{7}}{4}\.

These values are crucial for calculating other trigonometric functions and solving equations within the context of the unit circle. By understanding the relationship between these values and the points, it becomes easier to handle rotations and reflections as well. This foundational knowledge is essential for navigating more complex trigonometric concepts.