Problem 14 - 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. $$\cot (-x)=-\cot x$$

Step-by-Step Solution

Answer

We proved \(\cot(-x) = -\cot x\) using trigonometric identities for sine and cosine.

Step 1: Start with Cotangent Definition

To show that \(\cot (-x) = -\cot x\), start by writing the definition of cotangent in terms of sine and cosine: \(\cot(-x) = \frac{\cos(-x)}{\sin(-x)}\).

Step 2: Use Even-Odd Trigonometric Identities

Recall that \(\cos(-x) = \cos x\) because cosine is an even function, and \(\sin(-x) = -\sin x\) because sine is an odd function.

Step 3: Substitute and Simplify

Substitute the even-odd identities into the equation: \(\cot(-x) = \frac{\cos x}{-\sin x}\). This simplifies to \(-\frac{\cos x}{\sin x}\).

Step 4: Recognize the Simplified Expression

Recognize that the expression \(-\frac{\cos x}{\sin x}\) is equal to \(-\cot x\), completing the proof that \(\cot(-x) = -\cot x\).

Key Concepts

Even-Odd Identities

Trigonometric identities are fundamental in solving problems involving angles and their functions. Even and odd identities help in determining the behavior of trigonometric functions when the input value, or angle, is negated.

These identities rely on understanding whether a function is classified as even or odd, which greatly influences its symmetry.

• An even function satisfies the condition: \( f(-x) = f(x) \). It means the function reflects symmetrically across the y-axis.
• The cosine function \( \cos(x) \) is an even function.

Conversely, an odd function is defined by: \( f(-x) = -f(x) \). The graph of the function is symmetrical about the origin.

• The sine function \( \sin(x) \) is an odd function.
• The tangent function, which is derived from sine and cosine, is also odd.

These identities are crucial when transforming trigonometric expressions, especially when simplifying expressions that involve the negative of an angle.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is the reciprocal of the tangent function. It is essential to understand its behavior and properties when dealing with various mathematical problems, such as identities and transformations.

• Defined as: \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
• Cotangent is undefined when \( \sin(x) = 0 \), which occurs at integer multiples of \( \pi \).

In the context of even and odd identities, the cotangent function plays a role because it is composed of the even cosine function and the odd sine function. This composition makes the cotangent function itself an odd function, satisfying: \( \cot(-x) = -\cot(x) \). This property is used in many trigonometric transformations and proofs to simplify expressions involving negative angles.

Cosine Function

The cosine function is one of the primary trigonometric functions, symbolized as \( \cos(x) \). Understanding its characteristics is vital for grasping many trigonometric concepts.

• The cosine function is even, meaning it upholds the identity: \( \cos(-x) = \cos(x) \).
• This reflects the graph of \( \cos(x) \), which is symmetrically mirrored about the y-axis.

This even property is especially useful when dealing with expressions involving negative angles, as it allows for simplification by eliminating the negative sign. Additionally, cosine represents the x-coordinate on the unit circle when the angle is measured from the positive x-axis, influencing its applications in physics and engineering, particularly for oscillations and waves.

Sine Function

The sine function, expressed as \( \sin(x) \), is another fundamental trigonometric function. It displays unique characteristics that are important in various applications. Unlike the cosine function, the sine function is classified as odd.

• The defining property of the sine function is: \( \sin(-x) = -\sin(x) \).
• This odd property means that the sine function's graph is symmetric about the origin, as opposed to the y-axis.

Because of this odd nature, the sine function is used extensively in calculations involving oscillations and harmonic motion, where phase shifts and sign changes are relevant. Unlike cosine, sine represents the y-coordinate on the unit circle for a given angle, which is crucial for understanding rotational dynamics and other physical phenomena. By leveraging its odd property, simplifications in derivations and problem-solving become more apparent, especially when handling negative angle measures.