Problem 87 - Precalculus

Question

For Exercises \(87-90\), refer to the following: Set the calculator in parametric and radian modes and let $$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$ (TABLE CANNOT COPY) Set the window so that \(0 \leq \mathrm{T} \leq 2 \pi,\) step \(=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2\) and \(-2 \leq Y \leq 2 .\) To approximate the sine or cosine of a T value, use the \([\text { TRACE }]\) key, type in the T value, and read the corresponding coordinates from the screen. Approximate \(\cos \left(\frac{\pi}{3}\right),\) take 5 steps of \(\frac{\pi}{15}\) each, and read the \(x\) -coordinate.

Step-by-Step Solution

Answer

The x-coordinate approximates \( \cos\left(\frac{\pi}{3}\right) \), which is about 0.5.

Step 1: Set Calculator Mode

Ensure your calculator is in parametric mode and the angle is set to radians. This setting is crucial for dealing with trigonometric functions in parametric equations.

Step 2: Enter Parametric Equations

Enter the parametric equations for X and Y in your calculator: - For \ X: \( X_1 = \cos T \) - For \ Y: \( Y_1 = \sin T \).

Step 3: Configure Window Settings

Adjust the window settings on your calculator:- Set \( T \) from 0 to \( 2\pi \).- Set the step size to \( \frac{\pi}{15} \).- Set X from -2 to 2.- Set Y from -2 to 2. This creates a viewing window that fits the unit circle.

Step 4: Use TRACE function

Access the TRACE function on your calculator, which will help you find the coordinates for specific T values.

Step 5: Enter Target T Value

Use the TRACE function to enter the T value of interest, which is \( \frac{\pi}{3} \). The step size \( \frac{\pi}{15} \) allows for precise increments to accurately trace this position.

Step 6: Read the X-Coordinate

During TRACING, after entering \( T = \frac{\pi}{3} \), the calculator will display the coordinates (X, Y). The \( x \)-coordinate will approximate \( \cos \left(\frac{\pi}{3}\right) \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the context of parametric equations, they are used to describe circular motion and periodic phenomena.
The most common trigonometric functions are sine (\( ext{sin}\)) and cosine (\( ext{cos}\)). These functions are closely linked to the geometry of a circle. In parametric equations:

• Sine is associated with the y-coordinate.
• Cosine is associated with the x-coordinate.

These functions are periodic, which means they repeat their values in regular intervals. This characteristic is particularly useful in creating oscillating motions, like waves.
For example, in our problem, we use \(X_1 = \cos T \) and \(Y_1 = \sin T \), illustrating the parametric representation of a circle using trigonometric functions.

Unit Circle

The unit circle is a fundamental concept in trigonometry used to simplify the understanding of trigonometric functions. The idea is to have a circle with a radius of one unit centered at the origin of a plane. It helps to visualize how sine and cosine functions behave.

• The x-coordinate on the unit circle corresponds to the cosine value of an angle.
• The y-coordinate corresponds to the sine value of the same angle.

As the angle \(T\) changes from \(0\) to \(2\pi\), the point \(( \cos T , \sin T ) \) moves counterclockwise around the circumference of the circle.
In our exercise, this unit circle helps to visualize the movement and positions of different angles as you step through values of \(T\). Using this concept, you can intuitively understand why \(\cos(\pi/3)\) corresponds to a specific location on the circle.

Calculator Settings

Configuring your calculator correctly is pivotal when dealing with parametric equations, especially those involving radians. Here's how to set it up:

• Ensure your calculator is in parametric mode, allowing you to work with equations that define coordinates \((X, Y)\) in terms of a parameter, \(T\).
• Switch to radian mode because trigonometric functions in these exercises are usually expressed in radians.

Setting your calculator’s window settings properly is also crucial.

• Set \(T\) to range from \(0\) to \(2\pi\) to encompass a full circle trace.
• Set \(X\) and \(Y\) from \(-2\) to \(2\) to ensure visibility of the unit circle within the display.

Failing to adjust these settings might lead to incorrect representations of the graph or inaccurate results.

Graphing Techniques

Graphing parametric equations requires understanding some graphing techniques using your calculator. Once the calculator settings are correct, you can proceed to trace the graph.

• Using the TRACE function is an efficient way to determine the value of trigonometric functions at specific parameter values.
• Input the target \(T\) value, like \(\frac{\pi}{3}\) as in our example, to find the corresponding coordinates on the circle.
• The coordinates you see are essentially the cosine and sine at this specific angle.
  

Another crucial step is choosing an appropriate step size, such as \(\frac{\pi}{15}\), which ensures a smooth trace and accurate approximation. Graphical representation makes it much easier to visualize and estimate the value of trigonometric functions.