Problem 119 - Algebra and Trigonometry

Question

will help you prepare for the material covered in the next section. a. Find the exact value of \(\sin \left(\frac{\pi}{4}\right), \sin \left(-\frac{\pi}{4}\right), \sin \left(\frac{\pi}{3}\right),\) and \(\sin \left(-\frac{\pi}{3}\right) .\) Based on your results, can the sine function be an even function? Explain your answer. b. Find the exact value of \(\cos \left(\frac{\pi}{4}\right), \cos \left(-\frac{\pi}{4}\right), \cos \left(\frac{\pi}{3}\right),\) and \(\cos \left(-\frac{\pi}{3}\right) .\) Based on your results, can the cosine function be an odd function? Explain your answer.

Step-by-Step Solution

Answer

The sine function is odd, because \(f(-x) = -f(x)\) and the cosine function is even, because \(f(-x) = f(x)\). For the exact values, refer to steps 1 and 3.

Step 1: Calculate the value of sine functions

Use the unit circle or trigonometric values table to find the exact values of \(\sin \left(\frac{\pi}{4}\right)\), \(\sin \left(-\frac{\pi}{4}\right)\), \(\sin \left(\frac{\pi}{3}\right)\), and \(\sin \left(-\frac{\pi}{3}\right)\). The results are \(\frac{{\sqrt{2}}}{2}\), \(-\frac{{\sqrt{2}}}{2}\), \(\frac{{\sqrt{3}}}{2}\) and \(-\frac{{\sqrt{3}}}{2}\) respectively.

Step 2: Determine if the sine function is even or odd

An even function is symmetric around the y-axis, implying that \(f(x) = f(-x)\), while an odd function is symmetric around the origin, meaning that \(f(-x) = -f(x)\). By comparing the results from step 1, it can be seen that the sine function is not an even function as \(f(x) \neq f(-x)\), but it fulfills the characteristic of odd function as \(f(-x) = -f(x)\).

Step 3: Calculate the value of cosine functions

Use the unit circle or trigonometric values table to find the exact values of \(\cos \left(\frac{\pi}{4}\right)\), \(\cos \left(-\frac{\pi}{4}\right)\), \(\cos \left(\frac{\pi}{3}\right)\), and \(\cos \left(-\frac{\pi}{3}\right)\). The results are \(\frac{{\sqrt{2}}}{2}\), \(\frac{{\sqrt{2}}}{2}\), \(\frac{1}{2}\) and \(\frac{1}{2}\) respectively.

Step 4: Determine if the cosine function is even or odd

By comparing the results from step 3, it can be seen that the cosine function is an even function as \(f(x) = f(-x)\), but it does not fulfill the characteristic of odd function as \(f(-x) \neq -f(x)\).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that links the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. For instance, if you were to stand at the point of a right triangle where the angle is located, the side directly across from that point is the opposite side. The hypotenuse is the longest side of the triangle, opposite the right angle.

Here are a few key facts about sine:

• It is denoted as \( \sin(\theta) \), where \( \theta \) is the angle in question.
• The sine of an angle repeats its values every \( 2\pi \) radians or 360 degrees—the period of the sine function.
• This function is odd, which means \( \sin(-\theta) = -\sin(\theta) \).

When you calculate \( \sin(\frac{\pi}{4}) \), \( \sin(-\frac{\pi}{4}) \), \( \sin(\frac{\pi}{3}) \), and \( \sin(-\frac{\pi}{3}) \), you use either the unit circle or a trigonometric table to find exact values:

• \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \)
• \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \)
• \( \sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \)

This illustrates the odd nature of the sine function: changing the sign of the angle negates the value of the sine.

Cosine Function

The cosine function is another core trigonometric function, which correlates the angle in a right triangle with the ratio of the length of the adjacent side (next to the angle) to the hypotenuse. This function is crucial in understanding rotational dynamics and oscillations in physics.

Important properties of cosine include:

• It is represented as \( \cos(\theta) \), with \( \theta \) indicating the angle.
• The cosine function has a period of \( 2\pi \) radians as well, meaning that the function's values repeat every 360 degrees.
• Cosine is an even function, which means \( \cos(-\theta) = \cos(\theta) \).

When determining \( \cos(\frac{\pi}{4}) \), \( \cos(-\frac{\pi}{4}) \), \( \cos(\frac{\pi}{3}) \), and \( \cos(-\frac{\pi}{3}) \), the results will show the even trait:

• \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \cos(\frac{\pi}{3}) = \frac{1}{2} \)
• \( \cos(-\frac{\pi}{3}) = \frac{1}{2} \)

Thus, cosine maintains the same value regardless of the sign of the angle, confirming its even nature.

Even and Odd Functions

In mathematics, functions are categorized based on their symmetry properties around the y-axis and the origin. Identifying whether a function is even or odd plays a critical role in simplifying equations and solving integrals, particularly in trigonometry.

• An **even function** is symmetric about the y-axis. For any function \( f(x) \), it is even if \( f(-x) = f(x) \). Visually, this means that flipping the function’s graph over the y-axis yields the same graph.
• An **odd function** demonstrates symmetry with respect to the origin. A function is odd if \( f(-x) = -f(x) \). This implies a 180-degree rotation around the origin results in the same graph.

Trigonometric functions provide classic examples:

• The cosine function is an even function because flipping the angle's sign retains the function’s value, i.e., \( \cos(-\theta) = \cos(\theta) \).
• The sine function, on the other hand, is odd because when the angle's sign is reversed, the sine value is negated, i.e., \( \sin(-\theta) = -\sin(\theta) \).

Understanding these properties helps in evaluating these functions and predicting their behavior over various mathematical problems.