Problem 95 - Precalculus with Limits

Question

In Exercises 95 - 98, verify the identity. \( \cos(n\pi + \theta) = \left(-1\right)^n \cos \theta \), \( n \) is an integer.

Step-by-Step Solution

Answer

The verification of the identity \( \cos(n\pi + \theta) = (-1)^n \cos \theta \) can be seen directly through the positions of angles on the unit circle and understanding that the cosine function produces the x-coordinate of the point on the circle. The factor of \( (-1)^n \) accounts for the direction and number of half-turns (represented by \( n \)).

Step 1: Understand the position of angles on the unit circle

Any angle \( \theta \) can be defined as a point on the unit circle. If you add or subtract integer multiples of \( \pi \), you land on the same point on the unit circle. In other words, \( \cos(n\pi + \theta) \) and \( \cos(\theta) \) will land at the same point.

Step 2: Consider direction of rotation

The cosine function gives the x-coordinate of that point on the unit circle. Depending on the direction of rotation (anti-clockwise for positive angles and clockwise for negative angles) and number of turns, we can be going left or right on the x-axis. This is represented by \((-1)^n \), where \(n\) is the number of half-turns.

Step 3: Apply even-odd nature of integer

If \(n\) is an even integer, \( (-1)^n \) will be positive, which means that we don't change sides on the x-axis, hence \( \cos(n\pi + \theta) = \cos(\theta) \). If \(n\) is odd, \( (-1)^n \) will be negative, indicating a change of sides on the x-axis, making \( \cos(n\pi + \theta) = -\cos(\theta) \). However, this is the same as saying \( \cos(n\pi + \theta) = (-1)^n\cos(\theta)\) for all integer \(n\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that helps us understand the relationship between the angles and sides of a right-angled triangle. Specifically, in a right-angled triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

In the context of the unit circle, cosines are interpreted differently. Instead of relating to sides of a triangle, the cosine of an angle gives us the x-coordinate of a point on the unit circle. The unit circle is a circle with a radius of one, and the cosine function can take any real-number value of the angle \( \theta \) and output a corresponding x-coordinate point. This x-coordinate ranges between -1 and 1, corresponding to the full range of angles from 0 to \( 2\pi \) radians.

Unit Circle

The unit circle is a circle with a radius of exactly one unit, centered at the origin of a coordinate plane. This simple yet powerful concept is crucial for understanding trigonometric functions, because it maps angles to coordinates. Each point on the unit circle corresponds to an angle generated by the line connecting the point to the origin, compared to the positive x-axis.

Using the unit circle, the cosine function is visualized as the x-coordinate of the intersection point of the circle and a line forming the specified angle with the x-axis. Remember, as the angle increases, the point moves counter-clockwise around the circle, and the cosine value oscillates between -1 and 1.

Even-Odd Properties

Trigonometric functions showcase unique even-odd properties that can simplify the analysis of angles and their corresponding function values. Specifically for the cosine function, it is an 'even' function. This means that the function satisfies the property \( \cos(\theta) = \cos(-\theta) \). No matter if the angle is positive or negative, the cosine value of it will be the same, because it reflects over the y-axis on the unit circle.

When we look at the problem mentioned, we utilize these properties. For any integer \( n \), \( \cos(n\pi + \theta) \) will result in the cosine of the original angle \( \theta \) multiplied by \( (-1)^n \) to account for the direction of rotation around the unit circle, be it left or right, depending on whether \( n \) is even or odd.

Angle Addition Identities

Angle addition identities are a cornerstone of trigonometry, allowing us to express the sine or cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. The identity for the cosine of the sum of two angles is \( \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \).

These identities play a crucial role in solving complex trigonometric problems. The original problem \( \cos(n\pi + \theta) = (-1)^n \cos(\theta) \) is a specific application of the angle addition identity. By recognizing that adding multiples of \( \pi \) to an angle is essentially rotating around the unit circle by half-turns, we apply this identity in our step-by-step solution to verify the given trigonometric identity.