Problem 89
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 \(\sin \left(\frac{2 \pi}{3}\right)\) to four decimal places.
Step-by-Step Solution
Verified Answer
\(\sin\left(\frac{2\pi}{3}\right) \approx 0.8660\).
1Step 1: Setting Up the Calculator
First, ensure your calculator is set to "parametric" and "radian" modes. This is crucial to correctly input the parametric equations for \ \(X_1\) and \(Y_1\).
2Step 2: Inputting Parametric Equations
Input the equations \(X_1 = \cos(T)\) and \(Y_1 = \sin(T)\) into your calculator under the parametric mode. These will graph a unit circle as \(T\) varies.
3Step 3: Setting the Window Parameters
Adjust the window settings as follows: \(0 \leq T \leq 2\pi\), Step = \(\frac{\pi}{15}\), \(-2 \leq X \leq 2\), and \(-2 \leq Y \leq 2\). These settings ensure that your graph will fit nicely on the screen.
4Step 4: Tracing the Graph
Activate the \([\text{TRACE}]\) function on your calculator. This allows you to input a specified \(T\) value and find the corresponding \(X\) and \(Y\) coordinates.
5Step 5: Approximating \(\sin\left(\frac{2\pi}{3}\right)\)
While in the \([\text{TRACE}]\) function, input \(T = \frac{2\pi}{3}\). Read the \(Y\) coordinate displayed on the screen, which represents \(\sin\left(\frac{2\pi}{3}\right)\).
6Step 6: Finalizing the Approximation
The \(Y\) coordinate will display the approximation of \(\sin\left(\frac{2\pi}{3}\right)\) to four decimal places. Ensure you record this value accurately.
Key Concepts
Unit CircleParametric EquationsRadian Mode
Unit Circle
The unit circle is one of the most fundamental concepts in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. This simple construct allows us to easily determine the sine and cosine of various angles.
As we travel around the circle, the angle of rotation is measured in radians, starting from the positive x-axis. For any angle \(T\), the coordinates of the point on the unit circle are \((\cos(T), \sin(T))\). This means every point on the circle corresponds to the values of sine and cosine at a specific angle.
As we travel around the circle, the angle of rotation is measured in radians, starting from the positive x-axis. For any angle \(T\), the coordinates of the point on the unit circle are \((\cos(T), \sin(T))\). This means every point on the circle corresponds to the values of sine and cosine at a specific angle.
- \(\cos(T)\) gives the x-coordinate of a point on the circle.
- \(\sin(T)\) gives the y-coordinate of a point on the circle.
Parametric Equations
Parametric equations are a great way to represent curves and complex geometric shapes on a plane. Unlike traditional equations, where one variable is dependent on another, parametric equations express coordinates as functions of a third variable, typically \(T\) (a parameter).
With parametric equations, our focus is on \(X = \cos(T)\) and \(Y = \sin(T)\). As \(T\) varies from \(0\) to \(2\pi\), the point \((X, Y)\) traces the unit circle. This is because every coordinate on the unit circle is determined by some angle, \(T\), and the more values we take for \(T\), the more points we trace on the circle.
To visualize this on a graphing calculator, you input your parametric equations and observe how \((X, Y)\) changes as \(T\) changes. This is especially helpful in trigonometry, where seeing the physical representation of functions aids in understanding their properties.
With parametric equations, our focus is on \(X = \cos(T)\) and \(Y = \sin(T)\). As \(T\) varies from \(0\) to \(2\pi\), the point \((X, Y)\) traces the unit circle. This is because every coordinate on the unit circle is determined by some angle, \(T\), and the more values we take for \(T\), the more points we trace on the circle.
To visualize this on a graphing calculator, you input your parametric equations and observe how \((X, Y)\) changes as \(T\) changes. This is especially helpful in trigonometry, where seeing the physical representation of functions aids in understanding their properties.
Radian Mode
Radian mode is a way of measuring angles based on the radius of a circle. Instead of degrees, angles are measured as the length of the arc formed by that angle on a circle's circumference. This is why the whole circle is \(2\pi\) radians, as it corresponds to the total circumference of a unit circle.
Why use radians instead of degrees? Radians are more natural, especially when dealing with trigonometric functions, calculus and many mathematical concepts, since they directly relate to the geometry of circles itself.
When configuring a calculator for trigonometry tasks, it's vital to ensure it's set to radian mode. This is crucial because trigonometric functions like sine and cosine have different outputs depending if the angle is in radians or degrees. For instance, to find \(\sin\left(\frac{2\pi}{3}\right)\), your calculator should be in radian mode, which allows it to interpret \(\frac{2\pi}{3}\) as a radian measure on the unit circle.
Why use radians instead of degrees? Radians are more natural, especially when dealing with trigonometric functions, calculus and many mathematical concepts, since they directly relate to the geometry of circles itself.
When configuring a calculator for trigonometry tasks, it's vital to ensure it's set to radian mode. This is crucial because trigonometric functions like sine and cosine have different outputs depending if the angle is in radians or degrees. For instance, to find \(\sin\left(\frac{2\pi}{3}\right)\), your calculator should be in radian mode, which allows it to interpret \(\frac{2\pi}{3}\) as a radian measure on the unit circle.
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