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Chapter 2: Problem 21

Determine the period of each function. $$ y=5 \sec (2 x) $$

Short Answer

Expert verified
The period of the function is \(\pi\).

Step by step solution

01

Identify the General Form

The general form of a secant function is given by \(y = a \sec(bx)\), where \(a\) represents the amplitude and \(b\) affects the period.
02

Determine the Standard Period

The standard period of the secant function \(\sec(x)\) is \(2\pi\), because it is derived from the cosine function.
03

Calculate the Modified Period

To find the period of the function \(y = 5 \sec(2x)\), take the standard period of the secant function, \(2\pi\), and divide it by \(b\). In this case, \(b = 2\). Thus, the period is: \[ \text{Period} = \frac{2\pi}{b} = \frac{2\pi}{2} = \pi \]

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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 fundamental mathematical functions that relate the angles of a triangle to its side lengths. These functions are widely used in various fields such as engineering, physics, and mathematics. The main trigonometric functions include sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). All these functions are periodic, meaning they repeat their values in regular intervals. Each function has its own unique characteristics, including amplitude, period, and phase.
Secant Function
The secant function, abbreviated as sec, is defined as the reciprocal of the cosine function. The general form of the secant function can be written as: \(y = a \sec(bx)\). Here, 'a' represents the amplitude and 'b' affects the period of the function. The secant function has vertical asymptotes where the cosine function is zero because the secant is undefined at those points. As a result, the graph of the secant function has a pattern of upward and downward curves between these asymptotes.
Period of Function
The period of a function is the interval after which the function repeats its values. In other words, it measures the length of one complete cycle of the function. For standard trigonometric functions like \(\sec(x)\), \(\cos(x)\), and \(\sin(x)\), the period is \(2\pi\). When the function is modified to include a coefficient 'b' inside the argument, such as in \(\sec(bx)\), the period is adjusted by dividing the standard period by the absolute value of 'b'. Thus, for the function \(\sec(2x)\), the period becomes: \(\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{2} = \pi\).
Amplitude and Period
Amplitude and period are crucial properties of trigonometric functions. Amplitude refers to the maximum value a function reaches from its midpoint or average value. For functions like \(\sec(x)\), which are derived from the cosine function, the concept of amplitude can be a bit different as the secant function can stretch infinitely. The period is the interval required for the function to complete one full cycle. For \(y = 5 \sec(2x)\), the amplitude 'a' is 5, which scales the secant function vertically, and the period is \(\pi\), calculated as above. This understanding helps us grasp the oscillatory nature and repetition of trigonometric functions in various contexts.

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Most popular questions from this chapter

Let \(f(x)=\tan (x), g(x)=x+3,\) and \(h(x)=2 x .\) Find the following. $$ f(g(-3)) $$

Describe the graph of each function then graph the function between -2 and 2 using a graphing calculator or computer. $$ y=\sin \pi x-\cos \pi x $$

Write the equation of each curve in its final position. The graph of \(y=\tan (x)\) is shifted \(\pi / 2\) units to the left, shrunk by a factor of \(\frac{1}{2},\) then translated 5 units downward.

Evaluate without a calculator. Some of these expressions are undefined. a. \(\cos (\pi)\) b. \(\sin (3 \pi / 4)\) c. \(\tan (\pi / 3)\) d. \(\tan (\pi / 2)\) e. \(\sec (2 \pi / 3)\) f. \(\csc (\pi)\) g. \(\cot (5 \pi / 6)\) h. \(\sin (-\pi / 4)\)

Write the equation of each curve in its final position. The graph of \(y=\cot (x)\) is shifted \(\pi / 3\) units to the right, stretched by a factor of \(2,\) then translated 2 units downward
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