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Magazine Chapter 5: Problem 12
\(g(x)=\tan x \sec ^{2} x ; \quad[0, \pi / 4]\)
Short Answer
Expert verified
The function \(g(x)\) ranges from 0 to \(2\sqrt{2}\) on \([0, \pi/4]\).
Step by step solution
01
Understanding the Function
The function given is composed of trigonometric terms: \(g(x) = \tan x \sec^2 x\). We need to analyze the behavior of this function in the interval \([0, \pi/4]\). Remember, \(\sec x = \frac{1}{\cos x}\), hence \(\sec^2 x = \frac{1}{\cos^2 x}\).
02
Simplifying the Expression
Break down the function: \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec^2 x = \frac{1}{\cos^2 x}\). Therefore, \(g(x) = \tan x \cdot \sec^2 x = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos^2 x} = \frac{\sin x}{\cos^3 x}\).
03
Finding the Interval Values
Evaluate \(g(x)\) at the boundaries of the interval: 1. At \(x = 0\), \(g(0) = \frac{0}{1^3} = 0\).2. At \(x = \frac{\pi}{4}\), \(\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), so \(g\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}/2}{(\sqrt{2}/2)^3} = 2\sqrt{2}\).
04
Analyzing the Behavior within the Interval
The function \(g(x) = \frac{\sin x}{\cos^3 x}\) increases over the interval \([0, \pi/4]\), since \(\sin x\) slowly increases and \(\cos x\) slowly decreases. Hence, \(g(x)\) starts at 0 and ends at \(2\sqrt{2}\).
05
Conclusion
In conclusion, the function \(g(x)\) on the interval \([0, \pi/4]\) changes from 0 to \(2\sqrt{2}\). This is useful information if we need to analyze any integrals or further properties of \(g(x)\) in this domain.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Analysis
The core of analyzing a function, especially when dealing with trigonometric functions, is to break it down into its individual components. Here, the function in question is \(g(x) = \tan x \sec^2 x\). This function comprises two distinct trigonometric entities. Understanding these can make analyzing their combination much simpler.
- Firstly, the tangent function, denoted as \(\tan x\), is defined as \(\frac{\sin x}{\cos x}\). This relationship tells us how \(\tan x\) behaves depending on the sine and cosine values for any given angle \(x\).
- Next, the secant squared function, noted as \(\sec^2 x\), is equivalent to \(\frac{1}{\cos^2 x}\). Secant itself, \(\sec x\), is the reciprocal of cosine.
Trigonometric Identities
Trigonometric identities are powerful tools in simplifying and understanding trigonometric functions. In our function \(g(x) = \tan x \sec^2 x\), these identities help deconstruct and reorganize the expression for deeper analysis.
- One key identity applied here is \(\tan x = \frac{\sin x}{\cos x}\). This identity transforms the tangent function into a ratio of sine and cosine, revealing its nature and potential zero points.
- Another important identity involves the secant function: \(\sec x = \frac{1}{\cos x}\). Consequently, \(\sec^2 x = \frac{1}{\cos^2 x}\), which not only shows secant's relationship to cosine but also highlights the secant's positive increase as cosine values decrease.
Interval Evaluation
Evaluating a function within an interval involves examining how the function behaves at the boundaries and within that span of values. The focus for \(g(x) = \frac{\sin x}{\cos^3 x}\) is within the interval \([0, \pi/4]\). Here's how you perform such an evaluation:
- Check the function at the boundary points first. At \(x = 0\), we find that \(g(0) = \frac{0}{1^3} = 0\). This reveals that at the start of the interval, the function value is 0.
- At the other boundary, \(x = \pi/4\), both sine and cosine equal \(\frac{\sqrt{2}}{2}\). When substituting this into the function, \(g\left(\frac{\pi}{4}\right)\) computes to \(2\sqrt{2}\).
- To further understand the function within the interval, note its increasing nature: as sine increases and cosine decreases, the overall value of \(g(x)\) escalates.
