Problem 38 - Precalculus with Calculus Previews

Question

Justify the given statement with one of the properties of the trigonometric functions. $$ \sin (2 \pi / 3)=\sin (\pi / 3) $$

Step-by-Step Solution

Answer

The statement is justified using the sine of supplementary angles property: \( \sin(2\pi/3) = \sin(\pi - \pi/3) = \sin(\pi/3) \).

Step 1: Understanding the Problem

We need to justify the statement \( \sin (2 \pi / 3) = \sin (\pi / 3) \) using one of the trigonometric function properties. This involves identifying any symmetry, identities, or transformations that both angles \(2\pi/3\) and \(\pi/3\) might share.

Step 2: Recall the Property of Sine

The sine function has a property called the "sine of supplementary angles," which states that \( \sin(\pi - \theta) = \sin(\theta) \). This property takes advantage of the symmetry of the sine function around the line \( \theta = \pi/2 \).

Step 3: Apply the Supplementary Angle Property

Given \(2\pi/3\), we can express it as \(\pi - \pi/3\). Therefore, according to the property, \( \sin(2\pi/3) = \sin(\pi - \pi/3) = \sin(\pi/3) \). This justifies the given statement.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, describing how much an angle opens or closes from 0 to complete circle. It gives the ratio between the opposite side and the hypotenuse in a right triangle. Mathematically, it is denoted by \( \sin(\theta) \), where \( \theta \) is the angle. The sine function's values range between -1 and 1, corresponding to its peak and trough on a unit circle.

When graphed, the sine function forms a repetitive, wave-like pattern that oscillates every 360 degrees or \( 2\pi \) radians. This periodicity is due to its circular motion foundation.

• The peak occurs at \( \frac{\pi}{2} + 2k\pi \) and the trough at \( \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer.
• It crosses zero at integer multiples of \( \pi \).

Understanding sine's properties is crucial in various physics and engineering fields, involving waveforms, signal processing, and even simple pendulum motions.

Supplementary Angles

Supplementary angles are a pair of angles whose measures add up to \( \pi \) radians or 180 degrees. In the context of trigonometry, these angles highlight the sine function's symmetry. The property that connects the sine of supplementary angles is given by \( \sin(\pi - \theta) = \sin(\theta) \). This tells us how the sine values remain unchanged when you reflect across \( \pi/2 \).

Let's visualize it on a unit circle:

• A specific angle \( \theta \) is measured from the positive x-axis.
• The supplementary angle \( \pi - \theta \) would be the mirror of this angle across the vertical axis (y-axis).
• This symmetry means the y-coordinate, representing sine, is the same for both angles.

By utilizing this property, we can simplify and justify trigonometric expressions, much like in the problem of \( \sin(\frac{2\pi}{3}) = \sin(\frac{\pi}{3}) \).

Trigonometric Symmetry

Trigonometric symmetry refers to the symmetry characteristics shared by trigonometric functions on the unit circle. Specifically, the sine function exhibits symmetry around \( \pi/2 \) and \( 3\pi/2 \), thanks to supplementary angles as discussed earlier. This symmetry also extends into other functions and areas within trigonometry.

When considering symmetry:

• The sine function is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \), indicating symmetry about the origin.
• The cosine function, another trigonometric sibling, is even, so \( \cos(-\theta) = \cos(\theta) \), demonstrating symmetry about the y-axis.
• Thus, pairs of angles, such as \( 2\pi/3 \) and \( \pi/3 \), display trigonometric symmetry.

This symmetry makes it easier to understand the behavior of these functions over their cycle and aids in solving complex equations by simplifying expressions based on their mirrored values.