Problem 38
Question
Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{\sec \theta}{\csc \theta}$$
Step-by-Step Solution
Verified Answer
The simplified expression \(\frac{\sec \theta}{\csc \theta}\) is \(tan \theta\).
1Step 1: Identify the Fundamental Identities
First, we must identify the fundamental identities associated with secant (\(sec \theta\)) and cosecant (\(csc \theta\)). From trigonometry, we know that \( sec \theta = \frac{1}{cos \theta} \) and \( csc \theta = \frac{1}{sin \theta}.\)
2Step 2: Substitute the Identities in the Expression
Substitute the identities into the original expression. The given expression changes from \(\frac{sec \theta}{csc \theta}\) to \(\frac{\frac{1}{cos \theta}}{\frac{1}{sin \theta}}\)
3Step 3: Simplify the Expression
When dividing fractions, we can take the numerator (top) of the first fraction and multiply it by the reciprocal of the denominator (bottom) of the second one. So the new expression becomes \(\frac{1}{cos \theta} * sin \theta= sin \theta / cos \theta\)
4Step 4: Recognize the New Expression
The simplified expression \(sin \theta / cos \theta\) is another fundamental trigonometric identity. It is equal to \(tan \theta\).
Key Concepts
Fundamental Trigonometric IdentitiesGraphing UtilitySecant and CosecantReciprocal Trigonometric Functions
Fundamental Trigonometric Identities
Understanding the fundamental trigonometric identities is essential in simplifying trigonometric expressions and solving complex problems. These identities express trigonometric functions in terms of each other, which can often make calculations much easier.
One of the most commonly used sets of identities are the reciprocal identities. These state that the secant (\( \text{sec} \theta \)) is the reciprocal of the cosine, and the cosecant (\( \text{csc} \theta \) is the reciprocal of the sine. Specifically, \( \text{sec} \theta = \frac{1}{\text{cos} \theta} \) and \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \).
The power of these identities lies in their ability to transform a trigonometric expression into a form that can be much more manageable. For example, instead of adding or multiplying complex trigonometric functions, you can sometimes convert everything into sine and cosine. This simplification can then reveal opportunities to cancel out terms or combine them to make the expression easier to work with or solve.
One of the most commonly used sets of identities are the reciprocal identities. These state that the secant (\( \text{sec} \theta \)) is the reciprocal of the cosine, and the cosecant (\( \text{csc} \theta \) is the reciprocal of the sine. Specifically, \( \text{sec} \theta = \frac{1}{\text{cos} \theta} \) and \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \).
The power of these identities lies in their ability to transform a trigonometric expression into a form that can be much more manageable. For example, instead of adding or multiplying complex trigonometric functions, you can sometimes convert everything into sine and cosine. This simplification can then reveal opportunities to cancel out terms or combine them to make the expression easier to work with or solve.
Graphing Utility
A graphing utility is a powerful tool in verifying trigonometric identities and simplifying trigonometric expressions. When dealing with expressions like \(\frac{\sec \theta}{\csc \theta}\), it might not always be clear how the simplified form relates to the original expression on a graphical level.
To ensure that the simplification process hasn't altered the values of the expression within its domain, one can use a graphing utility to plot both the original and simplified expressions. The numerical table features of these utilities allows you to see a side-by-side comparison of the values for specific angles (\(\theta\)).
By comparing the outputs for a range of angles, the graphing utility can serve as a solid confirmation that the original and simplified expressions are indeed equivalent over their domain, thus verifying the identity numerically.
To ensure that the simplification process hasn't altered the values of the expression within its domain, one can use a graphing utility to plot both the original and simplified expressions. The numerical table features of these utilities allows you to see a side-by-side comparison of the values for specific angles (\(\theta\)).
By comparing the outputs for a range of angles, the graphing utility can serve as a solid confirmation that the original and simplified expressions are indeed equivalent over their domain, thus verifying the identity numerically.
Secant and Cosecant
Secant (\(\sec\)) and cosecant (\(\csc\)) are two trigonometric functions that are not as commonly encountered as their counterparts, sine and cosine. However, they are equally important, especially when it comes to simplifications and solving trigonometric equations.
Secant is the reciprocal of cosine, while cosecant is the reciprocal of sine. In their geometric interpretations, secant represents the hypotenuse divided by the adjacent side of the right-angled triangle, and cosecant is the hypotenuse over the opposite side.
Understanding these functions is crucial because they often show up in calculus in the form of integrals and derivatives of trigonometric functions, and also in polar or spherical coordinates. The challenge is not just to memorize them but to understand their geometrical meaning and how they relate to the other trigonometric functions, which can be a key in solving more complex problems.
Secant is the reciprocal of cosine, while cosecant is the reciprocal of sine. In their geometric interpretations, secant represents the hypotenuse divided by the adjacent side of the right-angled triangle, and cosecant is the hypotenuse over the opposite side.
Understanding these functions is crucial because they often show up in calculus in the form of integrals and derivatives of trigonometric functions, and also in polar or spherical coordinates. The challenge is not just to memorize them but to understand their geometrical meaning and how they relate to the other trigonometric functions, which can be a key in solving more complex problems.
Reciprocal Trigonometric Functions
In trigonometry, reciprocal functions play a central role in simplifying expressions. The three fundamental reciprocal trigonometric functions are secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)). These functions are respectively the reciprocals of cosine, sine, and tangent.
Specifically, we have the following relationships: \(\sec(\theta)=\frac{1}{\cos(\theta)}\), \(\csc(\theta)=\frac{1}{\sin(\theta)}\), and \(\cot(\theta)=\frac{1}{\tan(\theta)}\). These identities are not just mathematical curiosities; they serve as building blocks for simplifying complex trigonometric expressions, solving equations, and even proving other mathematical properties.
By applying these reciprocal relationships, one can manipulate an expression to reveal fundamental trigonometric ratios. For instance, in the original exercise, knowing that \(\sec\) and \(\csc\) are reciprocals of \(\cos\) and \(\sin\) respectively led us directly to the simplification of the expression to \(\tan \theta\), showcasing their practical utility.
Specifically, we have the following relationships: \(\sec(\theta)=\frac{1}{\cos(\theta)}\), \(\csc(\theta)=\frac{1}{\sin(\theta)}\), and \(\cot(\theta)=\frac{1}{\tan(\theta)}\). These identities are not just mathematical curiosities; they serve as building blocks for simplifying complex trigonometric expressions, solving equations, and even proving other mathematical properties.
By applying these reciprocal relationships, one can manipulate an expression to reveal fundamental trigonometric ratios. For instance, in the original exercise, knowing that \(\sec\) and \(\csc\) are reciprocals of \(\cos\) and \(\sin\) respectively led us directly to the simplification of the expression to \(\tan \theta\), showcasing their practical utility.
Other exercises in this chapter
Problem 38
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