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(i) Cos(n Pi+theta)=(-1)^(n)cos Theta

4 min read Jun 16, 2024
(i) Cos(n Pi+theta)=(-1)^(n)cos Theta

Understanding the Identity: cos(nπ + θ) = (-1)^n cos θ

This identity is a fundamental concept in trigonometry, allowing us to simplify trigonometric expressions involving angles of the form (nπ + θ) where 'n' is an integer.

Understanding the Derivation

The derivation of this identity relies on the following key trigonometric principles:

  • Periodicity of Cosine: The cosine function is periodic with a period of 2π. This means cos(x) = cos(x + 2πk) for any integer k.
  • Angle Addition Formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Let's break down the derivation:

  1. Expressing the angle: Start by expressing the angle (nπ + θ) as a sum of two angles: nπ and θ.

  2. Applying the Angle Addition Formula: Using the angle addition formula, we get:

    cos(nπ + θ) = cos(nπ)cos(θ) - sin(nπ)sin(θ)

  3. Evaluating Cosine and Sine of nπ:

    • cos(nπ) = (-1)^n (This is because cosine repeats every π with alternating values of 1 and -1)
    • sin(nπ) = 0 (Sine is zero at multiples of π)

  4. Substituting the values: Substitute the values of cos(nπ) and sin(nπ) into the equation:

    cos(nπ + θ) = (-1)^n cos(θ) - 0 * sin(θ)

  5. Simplifying the equation: The final result is:

    cos(nπ + θ) = (-1)^n cos θ

Applications of the Identity

This identity finds various applications in:

  • Simplifying trigonometric expressions: It allows us to simplify expressions involving angles of the form (nπ + θ), making it easier to work with them.
  • Solving trigonometric equations: The identity can be used to solve equations involving angles of the form (nπ + θ), as it helps to reduce the complexity of the equation.
  • Graphing trigonometric functions: Understanding this identity provides insights into the behavior of cosine function at different values of n, helping in visualizing and understanding its graph.

Example

Let's consider an example: Find the value of cos(7π/4) using the identity.

  1. Express 7π/4 in the form (nπ + θ): 7π/4 = 2π + (π/4). Here, n = 2 and θ = π/4.

  2. Apply the identity: cos(7π/4) = cos(2π + π/4) = (-1)^2 cos(π/4)

  3. Simplify: (-1)^2 = 1 and cos(π/4) = √2/2.

  4. Therefore, cos(7π/4) = √2/2.

Conclusion

The identity cos(nπ + θ) = (-1)^n cos θ is a powerful tool in trigonometry, offering a concise way to express and simplify trigonometric functions involving angles of the form (nπ + θ). It plays a significant role in understanding and manipulating trigonometric expressions, solving equations, and analyzing the behavior of trigonometric functions.

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