Problem 145 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 0 & 0.3 & 0.6 & 0.9 & 1.2 & 1.5 \\ \hline \cos \left(\frac{3 \pi}{2}-\theta\right) & & & & & & \\ \hline-\sin \theta & & & & & & \\ \hline \end{array}$$ (b) Make a conjecture about the relationship between \(\cos \left(\frac{3 \pi}{2}-\theta\right)\) and \(-\sin \theta\).

Step-by-Step Solution

Answer

Upon computing and comparing the function values for \( \cos \left(\frac{3 \pi}{2}-\theta\right) \) and \( -\sin \theta \), it appears that both functions produce the same values for the same \(\theta\) values, implying that \( \cos \left(\frac{3 \pi}{2}-\theta\right) = -\sin \theta \). However, this connection is a conjecture based on pattern observation and further mathematical reasoning or proof might be required to definitively validate this relationship.

Step 1: Compute the function values

Using a graphing utility, compute the values of \( \cos \left(\frac{3 \pi}{2}-\theta\right) \) and \( -\sin \theta \) for the given \(\theta\) values. Make sure to compute both functions for the same \(\theta\) values for comparison purposes.

Step 2: Fill in the table

Once the function values are computed, fill in the table. Every function value corresponds to a particular \(\theta\) value. Match the computed values to their corresponding \(\theta\) values.

Step 3: Observe the pattern

Upon filling the table, observe any pattern or relationship between the function values of \( \cos \left(\frac{3 \pi}{2}-\theta\right) \) and \( -\sin \theta \) for the same \(\theta\) values.

Step 4: Make a conjecture

Based on the observed pattern, make a conjecture about the relationship between \( \cos \left(\frac{3 \pi}{2}-\theta\right) \) and \( -\sin \theta \).

Key Concepts

Graphing Utility

If the terms graphing utility or graphing calculator are new to you, think of them as powerful tools that simplify complex calculations. These utilities can graph functions, calculate precise values, and even display comparisons side by side.
For this exercise, a graphing utility is used to compute and compare values of two trigonometric expressions at specific angles, \( \cos \left(\frac{3 \pi}{2}-\theta\right) \) and \( -\sin \theta \).
The utility provides a visual representation of functions, helping you quickly fill tables and observe patterns. You simply input a function, select angles, and hit calculate. The values reveal themselves with precision.
This removes manual error and enhances understanding when assessing how two trigonometric expressions relate.

Cosine Function

The cosine function, written as \( \cos \theta \), is essential in trigonometry. It assigns an angle \( \theta \) to a ratio, specifically the adjacent side over the hypotenuse in a right triangle.
The cosine function fluctuates between -1 and 1 and exhibits a periodic waveform. Understanding it helps visualize how changes in \( \theta \) affect the cosine value.

• Period: \( 2\pi \)
• Amplitude: 1
• Range: [-1, 1]

In this exercise, we examine \( \cos \left(\frac{3 \pi}{2}-\theta\right) \), a shifted version of the standard cosine. Graphing helps highlight its behavior and relevance when comparing function values.

Sine Function

The sine function, represented by \( \sin \theta \), is just as important as cosine in trigonometry. It describes a ratio between the opposite side and hypotenuse in a right triangle. Similar to cosine, sine values also range from -1 to 1 and form a wavelike pattern.
Key Characteristics Surface:

• Period: \( 2\pi \)
• Amplitude: 1
• Range: [-1, 1]

This exercise explores \( -\sin \theta \), which reflects the sine wave about the x-axis. Investigating this transformation shows how negative signs influence graph shapes and corresponding function values.

Function Values

Grasping function values is crucial to understanding trigonometric identities. They correspond to specific angles and offer insights about function behavior.
To complete the table in the exercise, you plug \( \theta \) into \( \cos \left(\frac{3 \pi}{2}-\theta\right) \) and \(-\sin \theta\). Comparing these outputs, we can identify if and how they relate.
Mathematically evaluating for given angles helps you make conjectures or predictions about relationships between different trigonometric functions. In particular, observing consistent matching or patterns in values can lead to insights into trigonometric identities, aiding in broader mathematical problem-solving.