Give A Rough Sketch Of The Following Function G(x) = Cotx = Cosx/sinx . Find Domain And Range.

Give A Rough Sketch Of The Following Function G(x) = Cotx = Cosx/sinx . Find Domain And Range.

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Introduction to the Cotangent Function

The cotangent function, denoted as G(x) = Cotx, is a fundamental trigonometric function that describes the ratio of the cosine to the sine of an angle x. Like other trigonometric functions, it exhibits periodicity, symmetry, and specific domain and range characteristics. Understanding these properties is essential for graphing the function accurately and applying it effectively in various mathematical contexts.

In this article, we will explore the definition of G(x) = cotx, analyze its domain and range, and provide a conceptual sketch of its graph, highlighting key features such as asymptotes, zeros, and periodic behavior.

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Definition and Basic Properties of G(x) = Cotx

Mathematical Definition

• G(x) = Cotx = Cosx / Sinx

where Sinx and Cosx are the sine and cosine functions, respectively.

Key Properties

• Periodicity: Cotx has a fundamental period of π (180°). This means the graph repeats every π units.
• Symmetry: The cotangent function is an odd function, satisfying G(-x) = -G(x). Its graph is symmetric with respect to the origin.
• Discontinuities: Points where Sinx = 0 lead to undefined values of cotx, producing vertical asymptotes.
• Range: G(x) can take all real values, from negative infinity to positive infinity, wherever it is defined.

Finding the Domain of G(x) = Cotx

Understanding the Definition Constraints

Points of Discontinuity

• Sinx = 0 at x = nπ, where n is any integer (n ∈ ℤ).
• Therefore, the points x = ..., -2π, -π, 0, π, 2π, 3π, ... are where G(x) is undefined.

Expressing the Domain

• The domain includes all real numbers except the points where Sinx = 0.
• Mathematically, the domain D is:

• D = ℝ \ {x | x = nπ, n ∈ ℤ}

This can be expressed as a union of open intervals:

• (-∞, -π), (-π, 0), (0, π), (π, 2π), (2π, 3π), ...

or more generally,

• For each integer n, the domain includes (nπ, (n+1)π), excluding the endpoints where the function is undefined.

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Analyzing the Range of G(x) = Cotx

Behavior of the Cotangent Function

• As x approaches points where Sinx = 0 from the left or right, cotx tends to ±∞.
• Between these asymptotes, cotx is continuous and monotonically decreasing from +∞ to -∞ over each interval.

Range Determination

• Since cotx approaches infinity or negative infinity near the vertical asymptotes and takes all real values in between, the range covers all real numbers.

Summary of Range

• Range: ℝ (all real numbers)

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Graphical Features of G(x) = Cotx

Vertical Asymptotes

• Occur at points where Sinx = 0, i.e., x = nπ.
• The graph approaches infinity or negative infinity as it nears these points.

Zeros of the Function

• Cotx = 0 where Cosx = 0 and Sinx ≠ 0.
• Cosx = 0 at x = (π/2) + nπ.
• Therefore, zeros of cotx occur at:

• x = (π/2) + nπ, where n ∈ ℤ.

Behavior Between Asymptotes

• On each interval between two asymptotes, the graph is a smooth, decreasing curve.
• It extends from +∞ to -∞ or vice versa, depending on the interval.

Periodicity and Symmetry

• The graph repeats every π.
• It is symmetric about the origin, reflecting its odd nature.

Rough Sketch of the Graph

Step-by-Step Visualization

1. Draw vertical dashed lines at x = nπ to indicate asymptotes.
2. Mark zeros at x = (π/2) + nπ.
3. On each interval between asymptotes:

• Start from +∞ near one asymptote.
• Pass through zero at the midpoint (where Cosx=0).
• Descend towards -∞ near the next asymptote.

Key Points to Remember

• The graph has a series of decreasing curves between asymptotes.
• It crosses the x-axis at x = (π/2) + nπ.
• Asymptotes are vertical lines at x = nπ.

Applications and Significance of G(x) = Cotx

Applications in Mathematics

• Solving trigonometric equations involving cotangent.
• Integration and differentiation in calculus.
• Analyzing periodic phenomena, such as wave behavior.

Real-World Contexts

• Engineering: signal processing, oscillations.
• Physics: describing angles of projection and wave interactions.
• Geometry: in problems involving right triangles and circle properties.

Conclusion

The cotangent function, G(x) = Cosx / Sinx, is a pivotal element of trigonometry with distinctive features. Its domain excludes points where Sinx = 0, i.e., multiples of π, leading to vertical asymptotes at these points. The range encompasses all real numbers due to the function's unbounded behavior between asymptotes. Its graph is characterized by periodic decreasing curves, zeros at specific points where Cosx = 0, and symmetry about the origin. Understanding these properties enables mathematicians and students to sketch the graph accurately and apply the cotangent function effectively in various analytical contexts.

This comprehensive overview provides the foundational knowledge necessary to visualize and interpret G(x) = cotx, highlighting its fundamental properties and behaviors.