Problem 12 - A Graphical Approach to Precalculus with Limits

Question

To show that sec(- \(x\) ) = sec \(x\) for all \(x\) in the domain, we begin by writing $$ \sec (-x)=\frac{1}{\cos (-x)} $$ and then use the fact that \(\cos (-x)=\cos x\) for all \(x\) to complete the argument. Use this method to prove each of the following. $$\csc (-x)=-\csc x$$

Step-by-Step Solution

Answer

\( \csc(-x) = -\csc(x) \) because \( \sin(-x) = -\sin(x) \) for all \( x \).

Step 1: Write the definition of cosecant

Start with expressing cosecant in terms of sine: \[\csc(-x) = \frac{1}{\sin(-x)}.\]

Step 2: Use the property of sine

Recall the property that sine is an odd function, which means \( \sin(-x) = -\sin(x) \) for all \( x \). Substitute this property into the previous expression:\[\csc(-x) = \frac{1}{-\sin(x)}.\]

Step 3: Simplify the expression

Simplifying, we find:\[\csc(-x) = -\frac{1}{\sin(x)}.\]Recognize that in terms of cosecant, this expression is:\[\csc(-x) = -\csc(x).\]

Step 4: Conclusion: Verify the property

From the steps above, we conclude that \( \csc(-x) = -\csc(x) \), demonstrating the property effectively for all \( x \) in the domain.

Key Concepts

Odd and Even Functions

In mathematics, functions possess unique traits defining their symmetry, known as odd and even properties. These properties help simplify calculations and verify identities.

• Even Functions: These functions exhibit symmetric behavior around the y-axis. This means that for any even function, if you input \(x\), the output is the same as when you input \(-x\). In mathematical terms, a function \(f(x)\) is even if \(f(-x) = f(x)\).
• Odd Functions: These functions have a rotational symmetry around the origin. An odd function satisfies the condition \(f(-x) = -f(x)\).

Both properties are crucial in trigonometry as they simplify solving identities. The sine function is an odd function, which helps in establishing relationships like \(\sin(-x) = -\sin(x)\). This is pivotal in proving identities for the cosecant function as shown in our original problem.

Cosecant Function

The cosecant function, often abbreviated as \(\csc(x)\), is one of the six main trigonometric functions. It is the reciprocal of the sine function. Understanding the cosecant function helps solve many trigonometric problems effortlessly.

• Definition: The cosecant of an angle \(\theta\) is the reciprocal of the sine of \(\theta\), expressed as \(\csc(\theta) = \frac{1}{\sin(\theta)}\).
• Behavior: Unlike sine, cosecant values are undefined when the sine is zero, as division by zero is undefined. Thus, \(\csc(\theta)\) is undefined for \(\theta = n\pi\), where \(n\) is an integer.

In the context of odd functions, the property \(\sin(-x) = -\sin(x)\) directly translates to \(\csc(-x) = -\csc(x)\) as demonstrated in the solution. This relationship emerges because the reciprocal aspect remains consistent even when multiplied by -1, displaying the symmetry of trigonometric functions.

Properties of Sine

The sine function boasts features that not only define its wave-like pattern but also establish foundational principles in trigonometry. Here's a closer look at its properties:

• Odd Function: As an odd function, sine satisfies \(\sin(-x) = -\sin(x)\). This is key to understanding trigonometric proofs and simplifying expressions involving sine and its related functions.
• Range and Domain: The sine function outputs values between \(-1\) and \(1\) and is defined for all real numbers, exhibiting periodic behavior with a period of \(2\pi\).
• Pythagorean Identity: One of the most famous identities involving sine is \(\sin^2(x) + \cos^2(x) = 1\). This is useful for deriving further trigonometric identities.

These properties facilitate the construction of trigonometric identities, such as proving the behavior of the cosecant function, as in the provided exercise.

Trigonometric Proofs

Trigonometric proofs are essential in establishing the validity of equations involving trigonometric functions. These proofs often rely on the definitions and properties of functions like sine, cosine, and their reciprocals.

• Identities: They are equations that are true for all values within the domain of the variable. Examples include Pythagorean identities, angle sum formulas, and reciprocal identities.
• Strategy: Most proofs involve either substitution or transformation by applying a known identity. Simplifying expressions using these identities often uncovers the truth of a given assertion.

In the provided solution, proving \(\csc(-x) = -\csc(x)\) started with substituting the expressed definition of cosecant using its relationship with sine and then applying the property of sine as an odd function. Such a clear structure is indispensable in understanding and constructing logical trigonometric arguments.