Problem 90 - 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 \(\cos \left(\frac{5 \pi}{4}\right)\) to four decimal places.

Step-by-Step Solution

Answer

\(-0.7071\)

Step 1: Set Calculator Mode

First, ensure your calculator is set to 'parametric' mode and use radians as the unit for measuring angles. This is essential for working with trigonometric functions like sine and cosine.

Step 2: Define Parametric Equations

Enter the parametric equations into your calculator: \[ X_{1}=\cos T \quad \text{and} \quad Y_{1}=\sin T \] These equations will plot the unit circle, where the x-coordinate represents \(\cos(T)\) and the y-coordinate represents \(\sin(T)\).

Step 3: Set Window Parameters

Adjust the window settings on your calculator to the following specifications: - Range for \(T\): \(0 \leq T \leq 2\pi\) - Step: \(\frac{\pi}{15}\) - X-Axis Range: \(-2 \leq X \leq 2\) - Y-Axis Range: \(-2 \leq Y \leq 2\) This ensures the entire unit circle is visible and accurately plotted.

Step 4: Use TRACE to Approximate

Press the [TRACE] key on your calculator. Enter \(T = \frac{5\pi}{4}\) using the numeric keypad. The calculator will show you the corresponding coordinates on the unit circle. The x-coordinate at this \(T\) value represents \(\cos\left(\frac{5\pi}{4}\right)\).

Step 5: Read and Record the Value

Read the x-coordinate, which is the value of \(\cos\left(\frac{5\pi}{4}\right)\). Ensure that the value displayed is to four decimal places. Typically for \(\frac{5\pi}{4}\), the cosine value should be around \(-0.7071\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of triangles. They are especially useful when dealing with right-angled triangles. The three main trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function represents a specific ratio:

• Sine (\( \sin \)): the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)): the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): the ratio of the opposite side to the adjacent side.

In the context of the unit circle, trigonometric functions are used to find the coordinates of a point on the circle. The x-coordinate is given by \( \cos(T) \) and the y-coordinate by \( \sin(T) \), where \( T \) is the angle in radians. Trigonometric functions are periodic, meaning their values repeat at regular intervals. For both sine and cosine, this period is \(2\pi\) radians. This periodicity allows you to calculate trigonometric values for angles that go beyond the usual \(0\) to \(2\pi\) range by cycling through the circle multiple times.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a useful tool for understanding the relationships between trigonometric functions and angles. Any point on this circle can be described using the angle \( T \) from the positive x-axis.
The unit circle simplifies calculations involving trigonometric functions because the radius is 1. This means the x-coordinate of a point on the circle is simply \( \cos(T) \) and the y-coordinate is \( \sin(T) \). These coordinates represent the cosine and sine of the angle \( T \), respectively.
Using the unit circle, we can easily determine trigonometric values such as \( \cos\left(\frac{5\pi}{4}\right)\), by identifying the coordinates at that particular angle. The unit circle also helps in visualizing how the sine and cosine functions vary with changes in angles, moving in a wave-like pattern as the angle progresses from \(0\) to \(2\pi\).

Calculator Settings

When working with trigonometric functions and parametric equations, calculator settings are important to get accurate results. Calculators must be set to 'parametric' mode when dealing with parametric equations, such as \( X_{1} = \cos(T) \) and \( Y_{1} = \sin(T) \).
On most calculators, parametric mode changes the calculator's graphing capabilities, allowing the graphing of sets of equations that define points in a plane based on a shared parameter, often time or \( T \).
Moreover, for trigonometric calculations involving radians, ensure the calculator is set to 'radians' mode instead of 'degrees'. Calculators can typically operate in either mode, so double-check this setting, as it affects the calculation outputs significantly. Forgetting to switch to radians when working with the unit circle or any radian-measured trigonometric problem will lead to incorrect results.

Radians Mode

Radians are a unit of angular measure used in many areas of mathematics. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of the circle itself. One full revolution around a circle encompasses \(2\pi\) radians, equivalent to 360 degrees.
When using a calculator for trigonometric functions in exercises involving the unit circle, it is crucial to work in radians mode. This setting ensures angles are interpreted correctly by the calculator, preserving the integrity of calculations involving periodic functions like sine and cosine.

• If you enter an angle without specifying, like \( \frac{5\pi}{4} \), the calculator will treat it as radians if it's set correctly.
• Radians are mathematically preferred for advanced calculations, as many mathematical properties and formulas conveniently use radian measure.

Understanding how to switch your calculator to radians mode is a fundamental skill when doing exercises involving trigonometric functions, ensuring your outputs align with theoretical expectations.