Problem 104 - Precalculus

Question

Determine whether each statement is true or false. \(\sin \theta=\sin (\theta+2 \pi n),\) where \(n\) is an integer.

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the Sine Periodicity

The sine function is periodic with a period of \(2\pi\). This means that \( \sin(\theta) = \sin(\theta + 2\pi) \) for any angle \( \theta \).

Step 2: Express General Form for sin(θ)

Because of the periodic nature of the sine function, we can consider \( \sin(\theta + 2\pi n) \), where \( n \) is any integer. Therefore, for any integer \( n \), \( \sin(\theta + 2\pi n) = \sin(\theta) \).

Step 3: Evaluate the Given Statement

The statement asks if \( \sin(\theta) = \sin(\theta + 2\pi n) \), where \( n \) is an integer. We know from the periodic property that this assertion holds true for all integers \( n \).

Key Concepts

Sine Function

The sine function is one of the most fundamental functions in trigonometry, often abbreviated as \( \sin \). It is defined for an angle \( \theta \) and represents the y-coordinate of the point on the unit circle corresponding to that angle. The sine function can take any real value and ranges between -1 and 1.
Understanding the sine function is crucial because it forms the basis for many applications in mathematics and physics. The function's graph is a smooth, continuous wave that repeats every \( 2\pi \) radians.
In practical terms, the behavior of the sine function can be observed in various natural phenomena, such as sound waves, light waves, and the oscillation of pendulums. Recognizing the typical sinusoidal pattern can help in analyzing systems that cycle periodically.

Periodicity

Periodicity refers to the property of certain functions to repeat their values in regular intervals known as periods. The sine function has a clear and defining period of \( 2\pi \) radians. This means that if you graph \( \sin(\theta) \) over an interval and then move by \( 2\pi \) radians in either direction, the pattern will perfectly repeat.
For the sine function, the concept of periodicity implies that:

• \( \sin(\theta) = \sin(\theta + 2\pi n) \)
• \( n \) is any integer
• This regular pattern allows for predictions and calculations across various disciplines

Periodicity makes trigonometric functions exceptionally useful in modeling cyclic phenomena like tides, seasons, and musical tones.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. These identities are essential tools in mathematics for simplifying expressions and solving trigonometric equations.
Some common trigonometric identities include the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and angle sum identities such as \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
These identities exploit the symmetries of the trigonometric functions and their periodic nature. For example, knowing that \( \sin(\theta) = \sin(\theta + 2\pi n) \) can simplify problems involving rotations and oscillations, as it allows variables to be reduced to within a single cycle of \( 0 \) to \( 2\pi \).
Understanding these identities can greatly aid in recognizing relationships between different trigonometric functions and angles, making complex problems more manageable.