Problem 63
Question
Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $$y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x}$$
Step-by-Step Solution
Verified Answer
The simplification of the given expression \(y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x}\) is \(\csc x\).
1Step 1: Plotting the function
The initial function \(y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x}\) can be simplified to \(y=\csc x - \cos^2x\csc x\). Use a graphing utility to sketch a graph of this function. Notice its pattern and behaviour.
2Step 2: Making a Conjecture
By observing the graph, one part is similar to a standard cosecant graph, \(\csc x\). Thus, can make the conjecture that the given expression simplifies to \(\csc x\).
3Step 3: Verifying the Identity Algebraically
To verify the identity algebraically, utilize the Pythagorean identity: \( \sin^{2}x + \cos^{2}x = 1 \). The expression can be rewritten as: \(\csc x - \cos^2x\csc x = \csc x(1 - \cos^{2} x)\)From here, rewrite \(1 - \cos^{2}x\) as \(\sin^{2}x\), getting \(\csc x\cdot \sin^{2}x\)Since \(\csc x\) is the reciprocal of \(\sin x\), \(\csc x\cdot \sin^{2}x = 1\cdot \sin x = \sin x\)The reciprocal of \(\sin x\) is \(\csc x\), thus validating the conjecture and the identity.
Key Concepts
Graphing UtilityPythagorean IdentityCosecant Function
Graphing Utility
A graphing utility is a tool, like a calculator or computer software, that can visually represent mathematical functions with graphs. These tools are extremely powerful when studying trigonometric functions because they help us see patterns, behaviors, and alternative expressions of these functions.
When working with trigonometric functions, plotting a graph gives us the ability to make conjectures about potential simplifications. For instance, when looking at the graph of the function \( y = \frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x} \), using a graphing utility can help reveal that this function may resemble the graph of the cosecant function, \( \csc x \).
When working with trigonometric functions, plotting a graph gives us the ability to make conjectures about potential simplifications. For instance, when looking at the graph of the function \( y = \frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x} \), using a graphing utility can help reveal that this function may resemble the graph of the cosecant function, \( \csc x \).
- Graphing utilities can visualize data quickly and accurately.
- They help in detecting periodic behavior and symmetries in trigonometric graphs.
- Such tools can aid in verifying algebraic simplifications through graphical comparison.
Pythagorean Identity
The Pythagorean Identity is one of the fundamental identities in trigonometry. It expresses the relationship between the sine and cosine functions through the equation \( \sin^{2}x + \cos^{2}x = 1 \). This identity is used frequently in simplifying and verifying trigonometric expressions.
In the given exercise, the Pythagorean Identity allows us to simplify the expression \( 1 - \cos^{2}x \) to \( \sin^{2}x \). This substitution is pivotal in reducing the complexity of the original function \( y=\csc x - \cos^2x\csc x \).
In the given exercise, the Pythagorean Identity allows us to simplify the expression \( 1 - \cos^{2}x \) to \( \sin^{2}x \). This substitution is pivotal in reducing the complexity of the original function \( y=\csc x - \cos^2x\csc x \).
- The identity helps in converting expressions between sine and cosine terms.
- It ensures simplifications that match common trigonometric function forms.
- Using the identity can transform an equation to make it easier to solve or verify.
Cosecant Function
The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. In mathematical terms, it is expressed as \( \csc x = \frac{1}{\sin x} \). Graphically, it shares similarities with the sine curve but features undefined points (vertical asymptotes) at integer multiples of \( \pi \), where the sine value becomes zero.
In the context of the exercise, recognizing \( \csc x \) within the expression \( y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x} \) can simplify the expression by identifying and canceling terms. The simplified expression \( \csc x(1 - \cos^{2} x) \) ultimately reduces to a more basic form due to the trigonometric identity substitutions.
In the context of the exercise, recognizing \( \csc x \) within the expression \( y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x} \) can simplify the expression by identifying and canceling terms. The simplified expression \( \csc x(1 - \cos^{2} x) \) ultimately reduces to a more basic form due to the trigonometric identity substitutions.
- \( \csc x \) is useful in a range of trigonometric transformations.
- The function is periodic with a period of \( 2\pi \).
- It reflects how reciprocal relationships can be used for algebraic manipulations.
Other exercises in this chapter
Problem 63
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Solve the multiple-angle equation. $$\tan 4 x=1$$
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Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sec u=\frac{7}{2}, \quad 0
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