Problem 64
Question
Perform indicated operation and simplify the result. $$\cos \beta(\sec \beta+\csc \beta)$$
Step-by-Step Solution
Verified Answer
The simplified result is \( 1 + \cot \beta \).
1Step 1: Rewrite Trigonometric Identities
Begin by expressing the given expression using basic trigonometric identities. Recall that \( \sec \beta = \frac{1}{\cos \beta} \) and \( \csc \beta = \frac{1}{\sin \beta} \). Substitute these into the given expression:\[ \cos \beta \left( \frac{1}{\cos \beta} + \frac{1}{\sin \beta} \right) \]
2Step 2: Distribute \( \cos \beta \) Inside the Parentheses
Distribute \( \cos \beta \) into the expression by multiplying it with each term inside the parentheses:\[ \cos \beta \cdot \frac{1}{\cos \beta} + \cos \beta \cdot \frac{1}{\sin \beta} \]
3Step 3: Simplify Each Term
Simplify each term separately: 1. \( \cos \beta \cdot \frac{1}{\cos \beta} = 1 \) 2. \( \cos \beta \cdot \frac{1}{\sin \beta} = \frac{\cos \beta}{\sin \beta} = \cot \beta \) Combine these results: \[ 1 + \cot \beta \]
4Step 4: Conclusion: Simplified Expression
The simplified result of the expression \( \cos \beta(\sec \beta + \csc \beta) \) is \( 1 + \cot \beta \).
Key Concepts
Simplifying Trigonometric ExpressionsTrigonometric FunctionsCotangent
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves transforming them into simpler forms by using known identities and basic algebraic manipulations. It allows us to understand and solve problems more effectively. In the original problem, the objective is to simplify the expression \( \cos \beta(\sec \beta + \csc \beta) \).
To start, it is crucial to recognize the identities that connect different trigonometric functions. For this exercise, the secant function \( \sec \beta \) is equivalent to \( \frac{1}{\cos \beta} \) and the cosecant function \( \csc \beta \) is equivalent to \( \frac{1}{\sin \beta} \).
This initial step converts the original expression into a form that's easier to handle by changing secondary functions into primary functions:
To start, it is crucial to recognize the identities that connect different trigonometric functions. For this exercise, the secant function \( \sec \beta \) is equivalent to \( \frac{1}{\cos \beta} \) and the cosecant function \( \csc \beta \) is equivalent to \( \frac{1}{\sin \beta} \).
This initial step converts the original expression into a form that's easier to handle by changing secondary functions into primary functions:
- \( \sec \beta \) becomes \( \frac{1}{\cos \beta} \)
- \( \csc \beta \) becomes \( \frac{1}{\sin \beta} \)
Trigonometric Functions
Trigonometric functions are essential tools in mathematics that relate the angles of a triangle to the lengths of its sides. They are fundamental in analyzing periodic phenomena, waves, and various fields of science and engineering.
In trigonometry, we often discuss six primary functions: sine \(\sin\), cosine \(\cos\), tangent \(\tan\), cotangent \(\cot\), secant \(\sec\), and cosecant \(\csc\). Each of these functions can be expressed in terms of a ratio of the sides of a right triangle, or using values from the unit circle, where:
As seen in the exercise, recognizing these traits allows for transforming and manipulating expressions effectively, making them more manageable and comprehensible.
In trigonometry, we often discuss six primary functions: sine \(\sin\), cosine \(\cos\), tangent \(\tan\), cotangent \(\cot\), secant \(\sec\), and cosecant \(\csc\). Each of these functions can be expressed in terms of a ratio of the sides of a right triangle, or using values from the unit circle, where:
- \( \cos \beta \) is the adjacent side over the hypotenuse.
- \( \sec \beta = \frac{1}{\cos \beta} \), representing the reciprocal of the cosine.
- \( \csc \beta = \frac{1}{\sin \beta} \), the reciprocal of the sine function.
As seen in the exercise, recognizing these traits allows for transforming and manipulating expressions effectively, making them more manageable and comprehensible.
Cotangent
The cotangent function, symbolized as \( \cot \), is an important trigonometric function. It is defined as the reciprocal of the tangent function. Mathematically, this is expressed as \( \cot \beta = \frac{1}{\tan \beta} \), or alternatively, \( \cot \beta = \frac{\cos \beta}{\sin \beta} \).
The cotangent provides insights into the relationship between the cosine and sine of an angle. It appears in various applications, particularly in calculus, signal processing, and solving trigonometric equations.
In the provided exercise, the cotangent function emerges naturally after simplifying the expression \( \cos \beta(\sec \beta + \csc \beta) \). Using the identity \( \frac{\cos \beta}{\sin \beta} \) for cotangent helps in attaining the final result, \( 1 + \cot \beta \).
This transformation showcases how understanding secondary trigonometric functions and their identities facilitates the simplification of mathematical expressions.
The cotangent provides insights into the relationship between the cosine and sine of an angle. It appears in various applications, particularly in calculus, signal processing, and solving trigonometric equations.
In the provided exercise, the cotangent function emerges naturally after simplifying the expression \( \cos \beta(\sec \beta + \csc \beta) \). Using the identity \( \frac{\cos \beta}{\sin \beta} \) for cotangent helps in attaining the final result, \( 1 + \cot \beta \).
This transformation showcases how understanding secondary trigonometric functions and their identities facilitates the simplification of mathematical expressions.
Other exercises in this chapter
Problem 64
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