Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) Use a graphing utility to complete the table. Round your results to four decimal places. $$\begin{array}{|l|l|l|l|l|l|}\hline \theta & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} \\\\\hline \sin \theta & & & & & \\\\\hline \cos \theta & & & & & \\\\\hline \tan \theta & & & & & \\\\\hline\end{array}$$ (b) Classify each of the three trigonometric functions as increasing or decreasing for the table values. (c) From the values in the table, verify that the tangent function is the quotient of the sine and cosine functions.

Step-by-Step Solution

Answer

Using a calculator, the missing values for sine, cosine and tangent for each angle were found. It was observed that sine and tan are increasing while cosine is decreasing over the given range. By dividing sine by cosine for each angle, it was confirmed that it equals the tangent of angle, hence verifying that the tangent is indeed the ratio of sine to cosine.

Step 1: Find the Trigonometric Values

Use your graphing utility or calculator to find the sine, cosine and tangent of the given angles. For instance, for \(0^\circ\), \(\sin(0^\circ) = 0\), \(\cos(0^\circ) = 1\) and \(\tan(0^\circ) = 0\). Repeat this for all the other angles.

Step 2: Analyze which is increasing or decreasing

Observe the values in the table. If the value is getting larger as we move from left to right, then it is increasing. If it's getting smaller, then it is decreasing. In our case, the sine and tangent values increase from \(0^\circ\) to \(80^\circ\), thus sine and tangent are increasing. The cosine values decrease from \(0^\circ\) to \(80^\circ\), which shows that cosine is decreasing.

Step 3: Verify the relationship between sine, cosine, and tangent

To verify that the tangent function values equal the quotient of the sine over the cosine, divide the sine value by the cosine value for each angle and see if it equals the tangent value. As an example, for \(20^\circ\), \(\tan(20^\circ) = \frac{\sin(20^\circ)}{\cos(20^\circ)}\). Repeat this for all other angles to confirm that it holds true in all cases.

Key Concepts

Utilizing a Graphing Utility

Graphing utilities, like graphing calculators or software, are invaluable tools in visualizing and understanding trigonometric functions. To start, let's discuss how to employ a graphing utility to complete an angle-based trigonometry table. After inputting the desired angles, such as 0°, 20°, and so on, the utility computes the sine, cosine, and tangent values for each angle. It is important to round the results as instructed, typically to four decimal places for precision while retaining clarity.

When using a graphing utility, it can display the waveforms representing the sine, cosine, and tangent functions. This not only aids in understanding the calculation of specific values but also provides a visual representation of how these functions behave and change over the interval of angles. By visualizing the curves, students can more easily grasp the periodic nature and symmetries of these functions, as well as the significance of key points where the functions intersect the axes.

Sine, Cosine, and Tangent Functions

In trigonometry, the sine, cosine, and tangent functions are fundamental, each relating an angle of a right-angled triangle to ratios of two sides of the triangle. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In contrast, the cosine relates the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.

The functions show cyclic patterns, which can be clearly observed in their graphical representations called waves. The sine and cosine functions are specifically known for their smooth wave-like graphs, while the tangent function, being a ratio of sine to cosine, results in a graph that exhibits both steep climbs and sharp descents, known as asymptotes.

Increasing and Decreasing Functions in Trigonometry

To classify trigonometric functions as increasing or decreasing over specific intervals, one can examine how their values change as the angle (\theta) increases. In the context of the table provided, we consider the behavior of the sine, cosine, and tangent functions as \theta progresses from 0° to 80°.

An increasing function is one in which the output value rises as the input (in this case, the angle) increases. Oppositely, a decreasing function sees the output value fall as the input ascends. So, within our considering interval, \(\theta = 0^{\text{°}}\) to \(\theta = 80^{\text{°}}\), the sine and tangent functions are increasing: their respective values grow larger with each increase in angle. Conversely, the cosine function is decreasing; its values diminish as the angle enlarges.

Understanding this behavior is essential because it helps predict the values of these functions without needing a calculator and gain deeper insights into their properties, such as their points of maximum and minimum, and intervals of growth and decay.