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Magazine Chapter 5: Problem 58
Find an equation using the cosecant function that has the same graph as \(y=\sec x\).
Short Answer
Expert verified
The equation is \(y = \csc(x - \frac{\pi}{2})\).
Step by step solution
01
Understand the Relationship Between Secant and Cosecant
The secant function, \(\sec x\), is the reciprocal of the cosine function, \(\cos x\). Similarly, the cosecant function, \(\csc x\), is the reciprocal of the sine function, \(\sin x\). To find an equation using \(\csc\) that matches \(y = \sec x\), we need to relate \(\cos x\) to \(\sin x\).
02
Convert the Trigonometric Function
Since \(\sec x = \frac{1}{\cos x}\), we want to express this in terms of \(\sin x\). We use the trigonometric identity \(\sin^2 x + \cos^2 x = 1\). This can be rewritten as \(\cos x = \sqrt{1 - \sin^2 x}\). However, substituting directly into \(\sec x\) will complicate finding a direct cosecant relation. We thus focus instead on a transformation that enables an exact match.
03
Match the Signs and Values
Secant and cosecant have vertical asymptotes where their respective cosine and sine values are zero. Since \(\sec x\) has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\) and \(\csc x\) has vertical asymptotes at \(x = k\pi\), we need a phase shift. Notice that a quarter-period phase shift modifies the graph. Specifically, translating sine more closely aligns its reciprocal values with \(\sec x\).
04
Determine the Shifted Cosecant Equation
To align both graphs, use \(y = \csc\left(x - \frac{\pi}{2}\right)\). This equation accounts for the phase shift moving the asymptotes of \(\sin x\) to align with those of \(\cos x\), hence replicating \(\sec x\).
05
Verify the Equation
Check that \(y = \csc(x - \frac{\pi}{2})\) matches \(y = \sec x\) for several points and asymptotes. The shifted graph should visually coincide with \(\sec x\), confirming the transformation is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosecant Function
The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. This means that \( \csc x = \frac{1}{\sin x} \). It is important to remember that wherever sine is zero, the cosecant function will have vertical asymptotes. Why? Because division by zero is undefined. This property is crucial when sketching or understanding the graph of the cosecant function.
- Vertical asymptotes occur at \( x = k\pi \), where \( k \) is an integer.
- These asymptotes split the graph into repeating cycles along the x-axis.
- Being a reciprocal function, \( \csc x \) exhibits similar wave-like properties but with significant differences from the sine graph.
Secant Function
The secant function, denoted as \( \sec x \), is the reciprocal of the cosine function, which translates to \( \sec x = \frac{1}{\cos x} \). Critical to grasp is the fact that \( \sec x \) will be undefined wherever \( \cos x \) is zero, leading to vertical asymptotes.
- These asymptotes appear at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
- Like \( \sec x \), it mirrors the period of \( \cos x \), having the same wavelength.
- The graph generally fulfills a pattern where secant waves crest when cosine troughs, producing a mirror-like alignment.
Phase Shift Transformation
In trigonometry, phase shift represents the horizontal movement of a graph along the x-axis. This is particularly useful when aligning graphs of different trigonometric functions. For example, a phase shift transformation can be leveraged to align the vertical asymptotes of \( \csc x \) with those of \( \sec x \).
- A positive phase shift moves the graph to the right.
- A negative phase shift moves the graph to the left.
- The amount of shift is represented by the term inside the trigonometric function.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions are inverse relationships to the primary trigonometric functions. These include secant, cosecant, and cotangent functions, expressed as inverses of sine, cosine, and tangent respectively.
- The reciprocal of sine is the cosecant (\( \csc x = \frac{1}{\sin x} \)).
- The reciprocal of cosine is the secant (\( \sec x = \frac{1}{\cos x} \)).
- The reciprocal of tangent is the cotangent (\( \cot x = \frac{1}{\tan x} \)).
