Problem 40 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \sin x \cot(-x) \)

Step-by-Step Solution

Answer

The simplified form of the given trigonometric expression \( \sin x \cot(-x) \) is \( -\cos x \).

Step 1: Apply the odd-even property

Use the odd-even property to simplify cotangent. As cotangent is an odd function, hence \( \cot(-x) = -\cot x \). So, our expression \( \sin x \cot(-x) \) turns into \( \sin x \cdot -\cot x \) which simplifies to \( -\sin x \cdot \cot x \).

Step 2: Express cotangent in terms of sine and cosine

The cotangent of angle x can be represented as the reciprocal of the tangent function i.e. \( \cot x = \frac{\cos x}{\sin x} \). Now, substitute \( \cot x \) with its equivalent form in \( -\sin x \cdot \cot x \) to get \( -\sin x \cdot \frac{\cos x}{\sin x} \).

Step 3: Simplify the expression

Simplify the expression further by canceling the common factor \( \sin x \) from numerator and denominator to left with \( -\cos x \).

Key Concepts

Understanding Cotangent

In trigonometry, cotangent is one of the six fundamental functions. It is defined as the reciprocal of the tangent function. So, mathematically, the cotangent of an angle \( x \) can be expressed as:

• \( \cot x = \frac{1}{\tan x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

This second expression shows that cotangent is the ratio of the cosine function to the sine function.
Cotangent behaves differently depending on the quadrant of the angle. It is positive in the first and third quadrants, and negative in the second and fourth. This is because of how sine and cosine change signs over different quadrants.
Understanding cotangent and its properties can make simplifying trigonometric expressions much easier, especially when combined with other trigonometric identities.

The Sine Function Explained

The sine function is one of the most basic and important trigonometric functions. Defined for any angle, it describes the y-coordinate of a point on the unit circle. The function can be expressed as:

• \( \sin x \)
• The opposite side over the hypotenuse in a right triangle

The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. Its range is from -1 to 1. Due to these properties, it is useful in modeling waves and oscillations.
The importance of the sine function also lies in its revelations about the angle's position within the unit circle. Understanding these properties assists in transformations and simplifications in various trigonometric problems.

Exploring Odd-Even Properties

In trigonometry, the odd-even properties help simplify expressions. An odd function satisfies \( f(-x) = -f(x) \), while an even function satisfies \( f(-x) = f(x) \). These properties make calculations easier and lead to more accurate transformations.
For the cotangent function, it is an odd function because \( \cot(-x) = -\cot(x) \). The sine function is also an odd function as \( \sin(-x) = -\sin(x) \).
These properties are particularly useful when dealing with trigonometric identities and symmetry, allowing you to transform and simplify expressions more easily. Recognizing odd and even functions at a glance can save time and effort during calculations and further understanding trigonometric graphs.

Basic Fundamental Trigonometric Identities

Fundamental identities form the foundation of trigonometry. These allow the transformation and simplification of trigonometric expressions. Key identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal Identities:
• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cot x = \frac{1}{\tan x} \)
• Quotient Identities:
• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

These identities enable the breakdown of complex expressions into simpler forms, facilitating easier evaluations. Memorizing these identities can immensely aid in solving various trigonometric problems and help connect different trigonometric concepts seamlessly.