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Chapter 4: Problem 5

Fill in the blanks. The period of \(y=\tan x\) is ___________.

Short Answer

Expert verified
The period of \(y=\tan x\) is \(\pi\).

Step by step solution

01

Identify the given function

Here, the given function is \(y = \tan x\). The function is a basic tangent function without any transformations.
02

Determine the period

The period of a basic tangent function is \(\pi\), i.e. it repeats its pattern after every \(\pi\) units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Functions
Trigonometric functions are a type of mathematical function that relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of angles and periodic phenomena seen in waves and circles. These functions include sine, cosine, and tangent, which are the most commonly used in various fields including physics, engineering, and navigation.
  • Sine Function (\( \sin x \)): Relates the opposite side to the hypotenuse in a right triangle. It starts from zero and rises to one at an angle of 90掳 (or \( \frac{\pi}{2} \) radians).
  • Cosine Function (\( \cos x \)): Relates the adjacent side to the hypotenuse. It starts from one, scaling down to zero at 90掳 and proceeding to negative values.
  • Tangent Function (\( \tan x \)): Expresses the ratio of sine to cosine. It can take values ranging from negative infinity to positive infinity and is undefined wherever cosine equals zero.
Trigonometric functions play a crucial role in periodic analysis through their repetitive oscillatory patterns.
Exploring Periodicity
Periodicity refers to the property of functions to repeat their values at regular intervals. This trait is quintessential in understanding cyclical processes and patterns in nature.

Trigonometric functions are a prime example of periodic functions. For instance, sine and cosine have a period of \( 2\pi \), meaning each function repeats its values every \( 2\pi \) units. Tangent functions, on the other hand, have shorter periods due to their vertical asymptotes.
  • Frequency of Cycles: The number of completed cycles per unit interval.
  • Phase Shift: The horizontal shift from the function's standard position to account for starting points not at zero.
Understanding periodicity is key to solving and predicting behaviors in oscillatory data, used in waves, vibrations, and circular motion.
Delving into the Basic Tangent Function
The basic tangent function, represented by \( y = \tan x \), is distinct due to its pattern and shape. Unlike sine and cosine, which oscillate, the tangent function grows towards infinity and drops back from negative infinity.The distinctive aspect of the tangent function is:
  • Vertical Asymptotes: These occur at odd multiples of \( \frac{\pi}{2} \) (e.g., \( \frac{\pi}{2}, \frac{3\pi}{2} \)
  • Period: The period of \( \tan x \) is \( \pi \), meaning it repeats its behavior after each \( \pi \) unit. This is notably shorter than the period of both sine and cosine functions due to the inclusion of asymptotes.
  • Range: Unlike sine and cosine, the range of the tangent function is all real numbers.
Thus, grasping the features of the tangent function helps in understanding trigonometric behaviors and their periodic nature.

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Most popular questions from this chapter

Data Analysis The table shows the average sales \(S\) (in millions of dollars) of an outerwear manufacturer for each month \(t,\) where \(t=1\) represents January. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Sales, } S & 13.46 & 11.15 & 8.00 & 4.85 & 2.54 & 1.70 \\\ \hline \end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { Sales, } S & 2.54 & 4.85 & 8.00 & 11.15 & 13.46 & 14.30 \\ \hline \end{array}$$ (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning. (d) Interpret the meaning of the model's amplitude in the context of the problem.

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