Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\cos (n \pi+\theta)=(-1)^{n} \cos \theta, \quad n \text { is an integer. }$$

Step-by-Step Solution

Answer

The identity \(\cos (n \pi+\theta)=(-1)^{n} \cos \theta\) is verified.

Step 1: Distribute the Function

First, we expand \(\cos (n \pi+\theta)\). Using the angle addition formula, we know this is equivalent to \(\cos(n \pi) \cos(\theta) - \sin(n \pi) \sin(\theta)\). Because \(\sin(n \pi) = 0\) for any integer \(n\), we can simplify this to \(\cos(n \pi) \cos(\theta)\).

Step 2: Evaluate Cosine at Multiples of Pi

The value of \(\cos(n \pi)\) alternates between 1 and -1 as \(n\) changes, which is equal to \((-1)^n\). Therefore, the function becomes \((-1)^{n} \cos(\theta)\).

Step 3: Compare Both Sides

Now, we have \((-1)^{n} \cos(\theta)\) on both sides of the equation, which verifies this is a valid identity for all integers \(n\).

Key Concepts

Angle Addition Formula

The angle addition formula is a key concept in trigonometry that allows us to simplify expressions involving the sum of two angles. Specifically for cosine, the angle addition formula is:

• \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \)

This formula helps break down complex trigonometric expressions, making them easier to solve or simplify.
In the given exercise, it is used to expand \( \cos(n\pi + \theta) \) into simpler parts. Using this formula is essential for verifying identities and solving trigonometric equations.
Understanding how to apply the angle addition formula accurately is crucial for anyone studying trigonometry, as it lays the groundwork for more advanced topics.

Integer Properties

In mathematics, integers are whole numbers that can be positive, negative, or zero. They play a crucial role in various mathematical operations and logic.
In the context of trigonometric identities, when dealing with multiples of \( \pi \), we often rely on integer properties. For example, \( n \) in the formula \( \cos(n \pi + \theta) \) represents any integer.
Understanding these integer properties is vital because they help determine the behavior of trigonometric functions. Particularly, they allow us to predict how the function \( \cos(n \pi) \) will behave, leading to the relation \( (-1)^n \), which is crucial for verifying the trigonometric identity in the problem.

• Integers make calculations predictable in cyclic or periodic patterns, such as those found in trigonometry.

Cosine Function

The cosine function, denoted as \( \cos \), is one of the fundamental trigonometric functions. It is used to calculate the horizontal coordinate of a point on the unit circle corresponding to a given angle.
The key property of the cosine function relevant to this exercise is its periodic nature. Specifically, \( \cos(n\pi) \) takes values of either 1 or -1, depending on whether \( n \) is an even or odd integer.

• \( \cos(0) = 1 \), \( \cos(\pi) = -1 \), and so on.

This alternation is captured neatly by the expression \( (-1)^n \). Understanding the behavior of the cosine function at multiples of \( \pi \) provides the backbone for verifying the given trigonometric identity. It's about recognizing these patterns and using them to simplify expressions and solve problems.

Trigonometric Functions

Trigonometric functions are a group of functions that relate angles in a right-angled triangle to the ratios of two side lengths. They are deeply intertwined with the geometry of the unit circle.
The main trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each offering a unique way to connect angles to side ratios. In trigonometry, these functions are not just limited to triangles but extend to oscillations, waves, and other phenomena.

• Trigonometric functions help in modeling periodic behavior due to their cyclic nature.
• They form the core of many mathematical identities and equations.

The exercise demonstrates the use of cosine, one of these essential functions, to verify an identity. Understanding these fundamental functions and their properties allows solving complex problems by simplifying them into more manageable forms.