Problem 143 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|} \hline \theta & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} \\\ \hline \sin \theta & & & & & \\ \hline \sin \left(180^{\circ}-\theta\right) & & & & & \\ \hline \end{array}$$ (b) Make a conjecture about the relationship between \(\sin \theta\) and \(\sin \left(180^{\circ}-\theta\right)\)

Step-by-Step Solution

Answer

For any given \(\theta\), \(\sin \theta\) is equal to \(\sin (180^{\circ} - \theta)\). This behaviour suggests that the sine function is symmetric about the \(90^{\circ}\) line in the coordinate plane.

Step 1: Point Plotting

Using a graphing utility, fill in the table as follows: \[\begin{{array}}{{|l|l|l|l|l|l|}}\hline\theta & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} \\hline\sin \theta & 0 & 0.342 & 0.643 & 0.866 & 0.984 \\hline\sin \left(180^{\circ}-\theta\right) & 0 & 0.342 & 0.643 & 0.866 & 0.984 \\hline\end{{array}}\]

Step 2: Observing the Pattern

By examining the completed table, you can see that for any given \(\theta\), \(\sin \theta\) is identical to \( \sin (180^{\circ} - \theta)\). This indicates a preserved symmetry in the sine function about the \(90^{\circ}\) line.

Step 3: Formulating the Conjecture

By considering the observed pattern, a conjecture about the relationship can be made: For any \(\theta\), \(\sin \theta = \sin (180^{\circ} - \theta)\). This represents a key property of the sine function in a unit circle and is consequential to its symmetry around the vertical axis.

Key Concepts

Graphing Utility Usage

Graphing utilities are essential tools when dealing with trigonometry, as they help visualize concepts and verify results. When you're asked to fill a table with values of a trigonometric function, a graphing calculator or software can provide accuracy and efficiency.

For instance, to complete a table for \(\theta\) and \(\sin \theta\), you can input the angle values into the graphing utility and obtain the corresponding sine values. It's crucial, however, to ensure that your calculator is set to the correct mode (degrees or radians) based on the given angle measurements. This simple step of using a graphical aid can help prevent calculation errors and support a deeper understanding of the trigonometric function's behavior.

Table Completion

Table completion is a method of organizing and summarizing values of functions, and it is particularly helpful in trigonometry for seeing patterns and relationships. To fill out a table, start by understanding what is required—here, you will calculate the sine of angles and their supplementary angles.

After each value is computed, whether by hand or graphing utility, you will begin to notice patterns. In our exercise, the values of \(\sin \theta\) and \(\sin (180^\circ - \theta)\) matched, leading to an insight about the sine function's symmetry. Tables are not only tools for calculation but also for discovery in mathematics.

Trigonometric Identities

Trigonometric identities are formulas that express relationships between trigonometric functions and angles. These identities are the backbone of solving complex trigonometry problems with ease. For example, one of the most fundamental identities is \(\sin(\theta) = \sin(180^\circ - \theta)\), stemming from the sine function's symmetry.

This identity exemplifies the principle that for acute angles, the sine of an angle is equal to the sine of its supplement. Understanding this can simplify calculations and proofs that would otherwise be more complex. Trigonometric identities reflect the periodic and symmetrical nature of these functions, making them invaluable for studying angle relationships.

Angle Relationships in Trigonometry

In trigonometry, angle relationships reveal how different angles relate to each other and their trigonometric functions. The symmetry in trigonometric functions, such as the sine function, shows that \(\sin \theta\) and \(\sin (180^\circ - \theta)\) are equal because they are essentially the same angles reflected over the y-axis on the unit circle.

This symmetric property is known as the 'co-function' identity and is vital for understanding the complementary and supplementary angles in trigonometry; they play a key role in simplifying expressions and solving problems. Recognizing these relationships helps us understand how angles and side lengths interact within different geometric shapes.