Problem 16 - Calculus with Concepts in Calculus

Question

Solve the inequality for \(x\) in \([0,2 \pi)\). Decide whether each of the following functions is even or odd. a. \(\sin x\) b. \(\cos x\) c. \(\tan x\) d. \(\cot x\) e. \(\sec x\) f. \(\csc x\)

Step-by-Step Solution

Answer

a. odd; b. even; c. odd; d. odd; e. even; f. odd.

Step 1: Understanding Even and Odd Functions

A function is called **even** if it satisfies the condition \( f(-x) = f(x) \) for all values of \( x \) in its domain. **Odd** functions satisfy the condition \( f(-x) = -f(x) \). Graphically, even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.

Step 2: Analyze \( \sin x \)

Check if \( \sin(-x) = -\sin(x) \). Since \( \sin(-x) = -\sin(x) \), \( \sin x \) is an **odd** function.

Step 3: Analyze \( \cos x \)

Check if \( \cos(-x) = \cos(x) \). Since \( \cos(-x) = \cos(x) \), \( \cos x \) is an **even** function.

Step 4: Analyze \( \tan x \)

Check if \( \tan(-x) = -\tan(x) \). Since \( \tan(-x) = -\tan(x) \), \( \tan x \) is an **odd** function.

Step 5: Analyze \( \cot x \)

Check if \( \cot(-x) = -\cot(x) \). Since \( \cot(-x) = -\cot(x) \), \( \cot x \) is an **odd** function.

Step 6: Analyze \( \sec x \)

Check if \( \sec(-x) = \sec(x) \). Since \( \sec(-x) = \sec(x) \), \( \sec x \) is an **even** function.

Step 7: Analyze \( \csc x \)

Check if \( \csc(-x) = -\csc(x) \). Since \( \csc(-x) = -\csc(x) \), \( \csc x \) is an **odd** function.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving equations and inequalities involving trigonometric functions. They serve as convenient shortcuts that allow you to simplify expressions by substituting equivalent expressions.
For example, the Pythagorean identity, one of the most common, states that \(\sin^2 x + \cos^2 x = 1\). This identity interrelates the square of sine and cosine functions and can be used to solve various equations.
Another fundamental pair of identities are the co-function identities. These showcase the relationship between trigonometric functions that are complementary to each other:

• \(\sin(\frac{\pi}{2} - x) = \cos x\)
• \(\cos(\frac{\pi}{2} - x) = \sin x\)

These identities allow switching between sine and cosine functions depending on the angle's complement.
Understanding and applying trigonometric identities is key to tackling complex trigonometric problems effectively. They enable you to transform expressions into more manageable forms, making problem-solving a much less daunting endeavor.

Function Symmetry

Function symmetry is a fundamental concept that helps us understand and classify functions based on their graphical representation.
Symmetry in functions often makes it easier to identify their characteristics and can simplify calculations.
There are two primary types of function symmetry:

• **Even functions**: A function \(f(x)\) is even if \(f(-x) = f(x)\) for every \(x\) in the domain. Graphically, even functions are symmetric with respect to the y-axis. A classic example of an even function is \(\cos x\), where \(\cos(-x) = \cos x\).
• **Odd functions**: A function \(f(x)\) is odd if \(f(-x) = -f(x)\) for every \(x\) in the domain. Odd functions exhibit symmetry with respect to the origin. Some noteworthy odd functions are \(\sin x\) and \(\tan x\), which have the properties \(\sin(-x) = -\sin(x)\) and \(\tan(-x) = -\tan(x)\), respectively.

Recognizing these symmetry properties is a powerful tool for both graphing and analyzing functions. It helps in simplifying problems, particularly those involving intervals or integrals of functions, by reducing the complexity of the calculations.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are pivotal in mathematics and describe relationships in right-angled triangles. Beyond their geometric interpretations, these functions have essential properties and are widely used in various fields like physics, engineering, and signal processing.
Let's break down some of the key trigonometric functions:

• **Sine** \(\sin x\) represents the ratio of the opposite side to the hypotenuse in a right triangle. It follows the identity \(\sin(-x) = -\sin x\), highlighting its odd nature.
• **Cosine** \(\cos x\) is the ratio of the adjacent side to the hypotenuse. Known for its even behavior, it satisfies \(\cos(-x) = \cos x\).
• **Tangent** \(\tan x\) is the ratio of the opposite side to the adjacent side, and is an odd function with \(\tan(-x) = -\tan x\).

These primary functions are the foundation upon which other trigonometric functions like secant \( \sec x \), cosecant \( \csc x \), and cotangent \( \cot x \) are built. They each provide unique relationships between angles and side lengths.
Mastery of these functions allows for effective problem-solving in trigonometry, aiding in tasks ranging from solving equations to modeling periodic phenomena.