Problem 87 Verify the identity. $$\cos (n... [FREE SOLUTION]

Chapter 5: Problem 87

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

Short Answer

Expert verified

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

Step by step solution

Rewrite the left-hand side in terms of cosine's periodicity

The frequency of the cosine function is $2\pi$, meaning $\cos\theta = \cos(\theta + 2n\pi)$ for any integer $n$. Using this, we can rewrite the left-hand side as $\cos(\theta + 2n \pi$. In our exercise, it will be transformed to $\cos(\theta + n\pi)$.

Deal with negative angles

Cosine function is an even function, meaning $\cos(-x) = \cos(x)$ for any x. So, for $n$ is odd, we write $\cos(-\pi) = \cos(\pi)$. Thus, $\cos(\theta + n\pi)$ can be re-written as $(-1)^{n}\cos(\theta)$

Verification

From the previous step, we have proved that left-hand side equals right-hand side. Thus, this verifies the given trigonometric identity for any integer $n$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Periodicity of Cosine Function

In trigonometry, understanding the periodicity of the cosine function is crucial. Periodicity refers to the repeating nature of the cosine wave. The cosine function is periodic with a period of $2\pi$. This means:

The function repeats its values every $2\pi$ units along the x-axis.
This can be expressed mathematically as $\cos(\theta) = \cos(\theta + 2n\pi)$, where $n$ is any integer.

In simpler terms, after moving along the x-axis by $2\pi$, the cosine function returns to its initial value.

This property is useful in verifying trigonometric identities. For instance, in our original exercise, we extended the argument by $n\pi$ rather than $2n\pi$. This transformation exemplifies how periodicity can simplify complex trigonometric problems.

Even Function Properties

An even function is symmetric about the y-axis, which indicates that the function tips over the y-axis without changing shape. Cosine is an even trigonometric function, meaning:

$\cos(-x) = \cos(x)$ for any angle $x$.
This property implies balance and symmetry regarding negative and positive angles.

Let's say we have to deal with negative angles like in the identity $\cos(n\pi + \theta)$. Using the even-function property, we know the cosine of an angle remains unchanged by negating it.

This characteristic aids in verifying identities, as seen in our step-by-step solution. When $n$ is odd, understanding that $\cos(-\pi) = \cos(\pi)$ simplifies calculations considerably.

Cosine of Integer Multiples of Pi

The cosine of integer multiples of $\pi$ holds unique values which play an essential role in trigonometric identities. Specifically, the values are:

$\cos(n\pi) = (-1)^n$ for integer $n$.
This results in $\cos(0) = 1$, $\cos(\pi) = -1$, $\cos(2\pi) = 1$, and so on.

These particular values cycle alternatively around -1 and 1 due to the nature of the cosine function.

In the original exercise, the identity $\cos(n \pi + \theta) = (-1)^n \cos(\theta)$ heavily relies on this concept. By understanding these values, calculations simplify, making trigonometric identities easier to verify. Whether $n$ is odd or even significantly affects the sign and outcome of the calculations.

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91影视

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

Short Answer

Step by step solution

Rewrite the left-hand side in terms of cosine's periodicity

Deal with negative angles

Verification

Key Concepts

Periodicity of Cosine Function

Even Function Properties

Cosine of Integer Multiples of Pi

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