/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Sketch the graph of each tangent... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 15

Sketch the graph of each tangent curve in the interval from 0 to 2\(\pi\) $$ y=\tan \theta $$

Short Answer

Expert verified
The graph of \( y = \tan \theta \) from 0 to 2\( \pi \) has an S shape, rising towards \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) to infinity and falling towards -infinity immediately after those points, with intercepts at \( \theta = 0, \pi, 2\pi \).

Step by step solution

01

Determine the Relevant Angles

Since the exercise requires plotting from 0 to 2\( \pi \), break that range into four sections: [0,\( \frac{\pi}{2} \)], [\( \frac{\pi}{2} \),\( \pi \)], [\( \pi \), \( \frac{3\pi}{2} \)], [ \( \frac{3\pi}{2} \), 2\( \pi \)].
02

Plot the Key Points

For the \( \tan(x) \) function, evaluate some key angles such as \( \theta = 0 \), \( \theta = \frac{\pi}{4} \), \( \theta = \frac{3\pi}{4} \), \( \theta = \pi \), \( \theta = \frac{5\pi}{4} \), \( \theta = \frac{7\pi}{4} \), and \( \theta = 2\pi \). This helps construct the graph based on where the function is rising, falling, or undefined.
03

Draw Asymptotes

The tangent function is undefined at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). Draw vertical lines at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). These are the asymptotes, where the function rises to positive or negative infinity.
04

Sketch the Curve

Based on the known tangent values and the asymptotes, sketch the curve in the four defined intervals. The tangent function increases as it approaches \( \frac{\pi}{2} \), decreases after passing \( \frac{\pi}{2} \) until \( \frac{3\pi}{2} \), and then rises again towards \( 2\pi \).

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions are sine, cosine, and tangent, each having unique properties and applications in mathematics and physics. The tangent function, in particular, is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.
  • Sine (\( ext{sin} \ heta)\) and Cosine (\( ext{cos} \ heta)\) Functions: These functions are periodic with a period of \(2\pi\), meaning they repeat their values every \(2\pi\) units.
  • Tangent (\( ext{tan} \ heta)\) Function: This function has a period of \(\pi\) and is undefined at angles where the cosine function is zero, leading to vertical asymptotes.
Tangent is particularly interesting because its graph possesses both increasing sections and undefined points, providing a rich landscape for exploration. Understanding the structure and behavior of trigonometric functions is essential for accurately sketching and analyzing their graphs.
Graphing Asymptotes
Graphing asymptotes involves identifying lines that a graph approaches but never touches. Asymptotes can be horizontal, vertical, or slant. In the context of the tangent function, vertical asymptotes occur where the function is undefined鈥攕pecifically, where the cosine of the angle is zero.
  • Vertical Asymptotes in \( y = \tan \theta \): These are found at the angles \( \frac{\pi}{2} + n\pi \), where \( n\) is an integer. In our interval \([0, 2\pi]\), they appear at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).
  • Purpose of Asymptotes: Asymptotes help to indicate the behavior of the function near these critical points, showing where the function values trend towards positive or negative infinity.
To sketch a tangent curve properly, it's crucial to accurately place these asymptotes on the graph, ensuring they are drawn as vertical lines and assist in shaping the function's graphical trajectory.
Intervals in Radians
In mathematics, particularly in trigonometry, radians provide a natural measure of angles, often offering more insight than degrees. Knowing how to work with radians is key to correctly plotting and interpreting trigonometric functions.
  • Definition of a Radian: One radian is the angle created when the radius is rotated around the circle such that the arc length equals the radius itself.
  • Conversion Insight: There are approximately \(57.2958\) degrees in a radian. Since a full circle is \(2\pi\) radians, half a circle is \(\pi\) radians.
Focusing on intervals in radians helps to understand the behavior of trigonometric graphs like the tangent curve. In the exercise solution, for instance, dividing \([0, 2\pi]\) into smaller intervals allows us to better visualize and sketch sections where the tangent function transitions through increasing, decreasing, and undefined phases.

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