Problem 34 - Precalculus Mathematics for Calculus

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 determined by each of the following. (a) \(-t\) (b) \(4 \pi+t\) (c) \(\pi-t\) (d) \(t-\pi\)

Step-by-Step Solution

Answer

\(-t\): \(\left(\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\), \(4\pi+t\): \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\), \(\pi-t\): \(\left(-\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\), \(t-\pi\): \(\left(-\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\).

Step 1: Understanding the Problem

We are given a point \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\) on the unit circle corresponding to angle \(t\). The goal is to find the terminal points for angles \(-t\), \(4\pi + t\), \(\pi - t\), and \(t - \pi\).

Step 1: Termination for \(-t\)

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

Step 2: Termination for \(4\pi + t\)

Adding \(4\pi\) to \(t\) completes two full revolutions around the unit circle, returning us to the original terminal point of \(t\). Thus, the terminal point is \(\left(\frac{3}{4}, \frac{\sqrt{7}}{4}\right)\).

Step 3: Termination for \(\pi - t\)

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

Step 4: Termination for \(t - \pi\)

The terminal point for \(t - \pi\) is obtained by reflecting the point for \(t\) over the origin, which changes the signs of both coordinates. Thus, the terminal point is \(\left(-\frac{3}{4}, -\frac{\sqrt{7}}{4}\right)\).

Key Concepts

Terminal Point

Understanding the concept of a terminal point is crucial when studying the unit circle. A terminal point corresponds to a specific angle on a circle that has a radius of one, hence the name "unit circle." Every angle has a point where its terminal side intersects the circle.
For a given angle, this point is described by coordinates ewline △

• the x-coordinate represents the cosine of the angle
• the y-coordinate stands for the sine of the angle

So, if you know the coordinates of a terminal point, you can determine the trigonometric function values for that angle. In our exercise, the initial terminal point is ewline ewline ewline\( \left( \frac{3}{4}, \frac{\sqrt{7}}{4} \right) \)connected to angle \(t\). The unit circle allows us to find these terminal points easily by moving around it in either direction, and reflecting them over certain axes when necessary.

Angle Reflection

When we reflect angles across axes in the unit circle, it affects the coordinates of the terminal point in simple, predictable ways. Depending on the kind of reflection, different coordinate changes occur:

• Reflecting over the x-axis changes the sign of the y-coordinate.
• Reflecting over the y-axis changes the sign of the x-coordinate.
• A double reflection or reflecting over the origin changes the sign of both coordinates.

Let's understand them with examples from the exercise:
- For angle ewline ewline-\(t\), the terminal point is reflected over the x-axis resulting in coordinates \(\left( \frac{3}{4}, -\frac{\sqrt{7}}{4} \right)\).
- For angle \(\pi - t\), reflection happens over the y-axis providing \(\left( -\frac{3}{4}, \frac{\sqrt{7}}{4} \right)\).Reflecting angles is an essential concept, which helps in exploring symmetrical properties of trigonometric functions.

Coordinate Transformation

Coordinate transformation within the unit circle involves altering the positions of points in a systematic way. In our case, it involves understanding periodic movements and reflections:

• Adding or subtracting from the angle causes the terminal point to rotate around the circle.
• Adding multiples of \(2\pi\) like \(4\pi\) essentially returns the point to the original position, as a complete rotation leads back to the start.

In the exercise:
- Adding \(4\pi\) to \(t\) results in the same terminal point because it represents a complete rotation with the same start and end points. \(\left( \frac{3}{4}, \frac{\sqrt{7}}{4} \right)\).- For angle \(t - \pi\), the transformation reflects the point over the origin, making both coordinates change signs resulting in \(\left( -\frac{3}{4}, -\frac{\sqrt{7}}{4} \right)\).These transformations help visualize how angles relate to each other locally and globally on the unit circle.

Trigonometric Functions

Trigonometric functions like sine and cosine are directly linked to coordinates on the unit circle. This deep connection helps in mapping these functions as seen on circular paths. Each point on the unit circle represents the solution to

△

• cosine corresponds to the x-coordinate,
• sine to the y-coordinate

of an angle. Understanding how terminal points reflect or transform strengthens our grasp of observations like how the value of sine becomes unpositive when reflected over the x-axis, or how cosine mirrors when reflected over the y-axis. These reflections and transformations do not change the radius of the circle, which remains 1, highlighting how trigonometric values cyclically repeat as angles increase or decrease. Identifying these patterns can make solving trigonometric problems much simpler.