Problem 62 - Calculus Single Variable

Question

A function \(f\) is said to have period \(p\) if there is a smallest positive number \(p\) such that \(f(x+p)=f(x)\) for all \(x\) in the domain of \(f\). Find the period of the function defined by the given expression. \(\sin (x)+\tan (x)\)

Step-by-Step Solution

Answer

The period of \( \sin(x) + \tan(x) \) is \( 2\pi \).

Step 1: Identify the Period of Sine

The standard sine function, \( \sin(x) \), has a period of \( 2\pi \). This means that \( \sin(x+2\pi) = \sin(x) \).

Step 2: Identify the Period of Tangent

The standard tangent function, \( \tan(x) \), has a period of \( \pi \). This means that \( \tan(x+\pi) = \tan(x) \).

Step 3: Find the Common Period

Since the expression involves both \( \sin(x) \) and \( \tan(x) \), the period of the combined function \( \sin(x) + \tan(x) \) will be the least common multiple of the periods of these functions, which are \( 2\pi \) for sine and \( \pi \) for tangent.

Step 4: Determine the Least Common Multiple

The least common multiple (LCM) of \( \pi \) and \( 2\pi \) is \( 2\pi \). This means that both functions repeat their values every \( 2\pi \).

Step 5: Conclude the Period of the Function

Therefore, the function \( \sin(x) + \tan(x) \) has a period of \( 2\pi \) because this is the smallest positive number such that the condition \( f(x+p) = f(x) \) holds for all \( x \) as both sine and tangent reset at \( 2\pi \).

Key Concepts

Periodicity

Periodicity refers to a function's behavior of repeating its values at regular intervals as you move along the x-axis.
For a function to be periodic, there should be a smallest positive number, called the period, such that when you add it to any x, the value of the function remains unchanged.

• This characteristic is formally described: a function \( f \) has period \( p \) if \( f(x + p) = f(x) \) for every \( x \) in its domain.
• If \( p \) is the smallest number for which this condition holds, \( p \) is known as the fundamental period.
• In real-world terms, periodic functions are like the cycles of a clock or seasons; they always return to their starting point after a set time.

Understanding periodicity is crucial for analyzing phenomena in mathematics and science, as it helps predict behaviors and outcomes.

Sine Function

The sine function is one of the fundamental trigonometric functions and plays a vital role in mathematics and physics.
It is frequently employed to describe oscillatory motions and waves.

• The defining property of the sine function is that \( \sin(x) \) repeats every \( 2\pi \) radians, meaning its period is \( 2\pi \).
• This pattern means that \( \sin(x + 2\pi) = \sin(x) \) for all values of \( x \).
• The sine function maps any angle to a value between -1 and 1, making it useful for describing cycles in an undulating pattern.

Sine's periodic nature is easily visualized in its graph, which forms continuous waves, peaking and then dipping back to follow the same path again and again.

Tangent Function

The tangent function is another crucial trigonometric function, often used to describe ratios in triangles and oscillations.
It is particularly useful in contexts where angle measures and slopes intersect.

• The key feature of the tangent function is its period of \( \pi \), meaning \( \tan(x + \pi) = \tan(x) \) for any angle \( x \).
• Unlike the sine and cosine functions, tangent does not have a maximum or minimum value and can take any real number.
• The graph of the tangent function is distinct from sine, with asymptotes and repeated patterns every \( \pi \) units.

Because of these properties, the tangent function is particularly useful in solving problems related to angles and slopes that repeat in smaller intervals than the sine or cosine functions.

Least Common Multiple

Finding the least common multiple (LCM) of numbers is essential when dealing with combined periodic functions.
LCM is the smallest positive number that is a multiple of two or more numbers.

• When determining the period of a combined function like \( \sin(x) + \tan(x) \), we find the LCM of their respective periods, \( 2\pi \) and \( \pi \).
• The LCM of \( 2\pi \) and \( \pi \) is \( 2\pi \), thus establishing the combined function's period.
• Using the LCM ensures that both functions reach their starting point simultaneously, allowing for the cycle to reset completely.

In mathematical problems, calculating the LCM can be leveraged to synchronize periodic events, just like finding a common beat in music or the timing of repeated events in engineering.